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few lines before : And besides, the 49th Prop. is not applicable to these two figures; because AH is not given in species, but is, by the step for which the citation is brought, proved to be given in species.
Prop. 83. in the Greek text, is neither well enunciated nor demonstrated. The 73d, which in this edition is put in place of it, is really the same, as will appear by considering (see Dr Gregory's edition) that A, B, F, E, in the Greek text, are four proportionals; and that the proposition is to show, that a, which has a given ratio to E, is to r, as B is to a straight line to which A has a given ratio; or, by inversion, that r is to A as a straight line to which A has a given ratio, is to B; that is, if the proportionals be placed in this order, viz. r, E, A, B, that the first r is to a, to which the second E has a given ratio, as a straight line to which the third A has a given ratio is to the fourth B; which is the enunciation of the 73d, and was thus changed, that it might be made like to that of Prop. 72. in this edition, which is the 82d in the Greek text: and the demonstration of Prop. 73. is the same with that of Prop. 72, only making use of Prop. 23. instead of Prop. 22. of Book 5. of the Elements.
PROP. LXXVII. This is put in place of Prop 79. in the Greek text, which is not a Datum, but a Theorem, premised as a Lemma to Prop. 80. in that text: And Prop. 79. is made cor. 1. to Prop. 77. in this edition. Cl. Hardy, in his edition of the Data, takes notice, that in Prop. 80. of the Greek text, the parallel KL in the figure of Prop. 77. in this edition, must meet the circumference, but does not demonstrate it, which is done here at the end of cor. 3. of Prop. 77. in the construction for finding a triangle similar to ABC.
PROP. LXXVIII. The demonstration of this, which is Prop. 80. in the Greek, is rendered a good deal shorter by help of Prop. 77.
PROP. LXXIX. LXXX. LXXXI. THESE are added to Euclid's Data, as propositions which are often useful in the solution of Problems.
This, which is Prop. 60. in the Greek text, is placed before the 83d and 84th, which in the Greek are the 58th and 59th, 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 86th Theorem, “ or proposition," seem 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 first two have no distinct meaning, and the third, which he says is the best, though it contains a true proposition, which is the 90th in this edition, has no connection 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 86th must be cancelled, and another put into its place : But the truth is, both the enunciation 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 propositions, which are the 88th and 90th 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 of Stanhope, and the other by Dr Mathew Stewart; to which I have added the demonstrations.
Though 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 ancient geometers never cite the data, and the moderns very rarely.
As that part of the composition of a problem which is its construction may not be so readily deduced from the analysis by beginners, for their sake the following example is given; in which the derivation of the several parts of the construction from the analysis is particularly shown, that they may be assisted to do the like in other problems.
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 sides and describe it.
The analysis of this is the same with the demonstration of the 87th 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 62d Prop. dat. therefore, from a point in FE, the perpendicular EG is drawn to FG, as the ratio of EF to EG is the ratio of the rectangle
B PD C
F G L O AB, BC to the parallelogram AC, by what is shown in 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 said 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 composition, 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 analysis: 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 said in the coustruction: And because it was shown that GO is to GL, as BC to BD, and that now the first three are found, the fourth BD is found by 12. 6. It was likewise shown, that LG is to FE, or GK, as DB to BA, and the first three are now found, and thereby the fourth BA. Make the angle ABC equal to EFG, and complete the parallelogram of which the sides are AB, BC, and the construction is finished; the rest of the composition contains the demonstration.
As the Propositions from the 13th to the 28th 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 show that they are equally useful with the other propositions, and from which it may easily be judged that many other problems depend upon these propositions.
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 rectangle contained 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 if a given straight line be taken from CD, the ratio of the remainder to EF is also given; therefore a the excess of the first a 24. dat. 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 AB above it, has a given ratio to EF; and consequently b the rectangle BA, A H B b 1. 6. AH, has a given ratio to the rectangle AB, EF, which last rectangle is given by the C G D hypothesis ; therefore the rectangle BA,
c 2. dat. AH is given, and BH the excess of its sides is given ; wherefore the sides AB, AH are givend: And because the ratios of KNML O d 85. dat. AB to CD, and of AH to EF are given, -CD and EFC are given.
The Composition.) SAVE A
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