PROP. LXXIX. LXXX. LXXXI.

These are added to Euclid's Data, as Propofitions which are often useful 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 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 tranflation by Mr. Bernard's, adds, that the 86. Theorem "or Propofition," feemed to be remarkably vitiated, but which could not be restored 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 distinct meaning, and the third which he fays is the beft, tho' it contains a true Propofition which is the 90. in this Edition, has no connexion in the leaft 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 fimple, 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 careleffness and ignorance of the Greek Editors should have been supplied, 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 Demonstrations.

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.

S that part of the Compofition of a Problem which is its Conftruction may not be fo readily deduced from the Anàlyfis 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 fhewn, that they may be assisted 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 excefs of the fquare 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 Demonstration 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 Analysis it is faid that the ratio of the rectangle AB, BC to the pa rallelogram 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

A

M

K

E

F GLO

HN

B PD C 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 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 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 GL, GH to FE, GH, that is) of the straight 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 square 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 square 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 square of BD, is to the fquare of BD, that is [8. 6.] as the fquare 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 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 square 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 straight 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 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 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 likewise fhewn 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 rest of the Compofition contains the Demonftration.

b. I. 6.

A

S the Propofitions from the 13. to the 28. may be thought by beginners to be lefs useful 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 shew that they are equally useful with the other Propofitions, and from which it may eafily be judged that many other Problems depend upon these Propofitions.

T

PROBLEM I.

10 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 rectangle con. tained by the firft and third is given.

Let AB be the first straight 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

b

a. 24. Dat. EF is given; therefore the excefs of the firft AB above a given ftraight line has a given ratio to the third EF. Let BH be that given ftraight line, therefore AH the excess of AB above it has a given ratio to EF; and confequently the rectangle BA, AH has a given ratio to the rectangle AB, EF, which last rectangle is given by the Hypothefis; E c. 2. Dat. therefore the rectangle BA, AH is given, and BH the excess of its fides is given; K d. 85. Dat. wherefore the fides AB, AH are given. and because the ratios of AB to CD, and of AH to EF are given; and EF are given.

c

The Compofition.

A

H B

C

G D

F

NML O

CD

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 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, fo GD to HB. to the given straight line

e

BH apply 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 ftraight lines required to be found. and by making as LK to KM, so AB to DC, DC will be the fecond. and lastly, make as KM to KN, fo CG to EF, and EF is the third.

For as AB to CD, so is HB to GD, each of these ratios being the fame with the ratio of LK to KM; therefore f AH is to CG, f. 19. 5. 1 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 g. 1. 6. re&angle 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 ftraight line GD being taken, the remainder CG has to EF the given ratio of KM to KN. Q. E. D.

PROB. II.

O 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 squares of the first and third is given.

Let AB be the first straight 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 first AB above a given 3. 24. Dat. 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 species; wherefore the angle BED b. 44. Dat. is given. let AE which is given in magnitude be given also in pofi

b

tion, and the ftraight 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. I. of AD is given, therefore the ftraight line AD is given in magni

e

tude; and it is also given in pofition, because from the given e. 34. Dat. point A it is drawn to the straight line ED given in pofition. therefore the point D in which the two ftraight lines AD, ED given in

G. g