PROP. LXXIX. LXXX. LXXXI.

These are added to Euclid's Data, as Propofitions which are often ufcful in the folution 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 thefe 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 fupplied the defects of the Greek text of the Data in innumerable places from feveral Manuscripts, and corrected Cl. Hardy's translation by Mr. Bernard's, adds, that the 86. Thcorem" or Propofition," feemed to be remarkably vitiated, but which could not be reftored by help of the Manufcripts; then he gives three different tranflations of it in Latin, according to which he thinks it may be read; the two first have no diftin&t meaning, and the third which he says is the best, tho' it contains a true Propofition which is the 90. in this Edition, has no connexion in the leaft with the Creek 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 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 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, fhould have been fupplied, not by changing Prop. 86. which is both entire and neceffary, but by adding the two Propofitions which are the 88. and 90. 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 Demonftrations.

Tho' 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.

AS

As

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 gi ven in which the derivation of the several parts of the Construction from the Analysis is particularly fhewn, that they may be assisted to do the like in other Problems.

PROBLE M.

Having given the magnitude of a parallelogram, the angle of which ABC is given, and alfo the excefs of the fquare of its fide BC above the fquare of the fide AB; To find its fides and defcribe it.

The Analyfis of this is the fame with the Demonftration of the 87. 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 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 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

AC by what is fhewn 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 Analysis 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 straight line DB to BA, is the fame with the ratio (of the rectangle

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 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 spaces 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 square of BD; and, by Compofition, four times the fquare of FE together with the fquare of GL is to the square 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 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 square of ML is to the fquare of GL, as the square of the ftraight line made up of BC and CD is to the fquare 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 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 faid to be given in the Analysis. 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 square 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 Construction. and because it was fhewn 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 fhewn that LG is to FE, or GK, as DB to BA, and the three first 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 Demonftration.

AS

As

S the Propofitions from the 13. to the 28. may be thought by beginners to be lefs ufeful than the reft, because they cannot fo readily fee how they are to be made ufe of in the folution of Problems; on this account the two following Problems are added, to fhew that they are equally useful with the other Propositions, and from which it may easily be judged that many other Problems depend upon thefe Propofitions.

PROBLEM I.

To find three straight lines fuch, that the ratio of the first to the fecond is given; and if a given straight line be taken from the fecond, the ratio of the remainder to the third is given; also the rectangle contained by the firft and third is given.

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 a. 24. Dat. EF is given; therefore the excess of the first AB above a given ftraight line has a given ratio to the third EF. Let BH be that gi ven ftraight line, therefore AH the excess of AB above it has a given ratio to EF; and b. 1. 6. confequently the rectangle BA, AH has a given ratio to the rectangle AB, EF, which laft rectangle is given by the Hypothefis; c. 2. Dat. therefore the rectangle BA, AH is given, and BH the excess of its fides is given; where

A

C

H B

G D

E F

K NML 0

d. 85. Dat.fore the fides AB, AH are given. and becaufe the ratios of AB to CD, and of AH to EF are given; CD and EF are given.

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; alfo, let the given 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

by making as KM to KL, fo GD to HB. to the given straight line

BH apply a rectangle equal to LK, KO exceeding by a fquare, and e. 19. 6. 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, so AB to DC, DC will be the fecond. and laftly, make as KM to KN, fo CG to EF, and EF is the third.

For as AB to CD, fo is HB to GD, the fame with the ratio of LK to KM;

as (AB to CD, that is, as) LK to KM;

each of these ratios being therefore f AH is to CG, f. 19. s 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 g. 1. 6. rectangle LK, KO to the rectangle KN, KO. and, by the Conftruction, the rectangle BA, AH is equal to LK, KO, therefore the h. 14. 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. ftraight line GD being taken, the remainder CG has to EF the gi ven ratio of KM to KN. Q. E. D.

PRO B. II.

To find three ftraight lines fuch, that the ratio of the first to the fecond is given; and if a given ftraight line be taken from the fecond, the ratio of the remainder to the third is given; alfo the fum of the fquares of the -first 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 straight line be taken from BC, the ratio of the remainder to BD is given; therefore the excess of the firft AB above a given a. 24. Dať, ftraight 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. Dati is given. let AE which is given in magnitude be given alfo in poition, and the straight line ED will be given in pofition. join AD, c. 32. Dat ̧ and because the fum of the fquares of AB, BD, that is, the fquare d. 47. 1. of AD is given, therefore the ftraight line AD is given in magnitude; and it is also given in position, because from the given e. 34. Ďa 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