BOOK V. occupied with the vulgar notion of proportion, which, though obfcure, is familiar, and attracts the attention whenever proportionals are mentioned. Now, the writer who would demonftrate this propofition as it ftands, must overcome all these difficulties. He must fhew, that the magnitudes affumed in the hy pothefis, are poffible, and muft place the dependence of every ftep of the demonftration upon the 5th definition, in the cleareft point of view, so as to call the reader's attention from the vulgar to the accurate notion of proportionals, contained in that definition, and to compel his mind, upon ad nitting that definition, to admit the feveral conclufions drawn from it. Inftead of this, Euclid affumes the poffible exiftence of four proportionals, a thing inconfiftent with accuracy, and quite contrary to his method in all fimilar cafes; and he draws his conclusions as implicitly as if his readers had been perfectly acquainted with the fubject: It is therefore evident, that this is not the first time that he has ufed the fifth definition, and that the place in which this propofition ftands at prefent, is not the place which he had allotted to it. Befides, it is placed at a diftance from all the propofitions about ratios, in the midft of propofitions about multiples, and that without any neceflity, for the first reference to it is in the 22d, and therefore the proper place of it is between the 19th and 20th: and the propofition which is made a corollary to it in the Greek, appears to have been originally in the fame place, for it is not used before the 20th; though, if it had been given before, feveral of the for. mer demonstrations could have been fhortened by it. It would be proper, therefore, to restore these propofitions to their original fituation, but this cannot be conveniently done with the fourth, because geometers very often cite the propofitions of the fifth book; and therefore the order is ftill preferved, but the demonstration is rendered as full, and its dependence on the fifth definition as plain as poffible.

Likewife, particular equimultiples are taken of the fecond and fourth, because it is thought, that the reader will better perceive the force of the demonftration, when he fees the equimultiples of the fecond and fourth, which contain them the fame number of times that the equimultiples of the first and third contain them. And the fame particularity is used in all the following propofitions, where the demonftrations require equimultiples of the fecond and fourth to be taken. But the demonftrations could have been made in the fame way, though any equimultiples had been taken of them and Euclid's demonftrations could have been conducted the fame way, if he had taken equimultiples in the manner that is now done, and they would have been as general as when general equimultiples

are

are taken, for the generality of the demonftration depends wholly Book V. upon the general equimultiples of the first and third.

The 7th would have answered the purpose better, if it had been made more general: Thus, "If the first of four magnitudes be equal to the third, and the fecond to the fourth; the first is to the fecond, as the third to the fourth :" and we find that Euclid understood it in this general fenfe, for the first application of it which he makes, is to four magnitudes in the 15th of this book. But however it be expreffed, it is evident, from the nature of equal things, that in all cafes relating to their magnitude only, one of them may be substituted for another.

There are fix propofitions added to this book, of which the first five are placed between the 19th and 20th, because fome of them are neceffary to the following propofitions. It was fhewn, in the notes on the 4th, that this is the proper place of the fe cond of them, and it is evidently the proper place of the fifth; and it was thought to be the best way to place them altogether.

The fixth of them is one of thofe relating to compound ratio, which Dr Simfon placed at the end of this book, and it is thought to be fufficient for fhewing, that what is said of com pound ratio by geometers, may be understood from the 22d and 23d of this book.

B OO K VI.

THE zd definition is made more general and accurate, than Book VI.

TH

in the former editions of Euclid.

In the enunciation of the 7th propofition, the limitation is altered, fo as to agree with what is proved in the demonftration to be the case, when the triangles are not equiangular, by which means a diftinction of cafes is avoided. But it is as eafily known, when the prefent exception does not take place, as when the former does not take place.

Dr. Simfon obferves, that the 18th is vitiated, for it is only demonftrated of quadrilaterals: and there certainly is but one cafe mentioned in the Greek, which is conftructed and demonftrated in a manner very different from. Euclid's. But when a perfon, acquainted with Euclid's manner, attempts to restore it, he naturally falls into two cafes, that of a triangle, and that of a quadrilateral; for in the Greek, the cafe of the triangle is first conftructed and demonftrated, and then the quadrilateral is conftructed from it. These two cafes are now diftinguished; and from them it appears how to extend it to figures of five or more fides, as Dr Simfon has done.

Рpa

Many

Book VI.

Many complaints have been made of the 27th, 28th, and 19th; but they are not in reality more difficult than the 25th, of which. there are no complaints. The enunciations, however, are very obfcurely expreffed; and the conftructions of the 28th and 29th are not fufficiently full, for it is taken for granted, that the reader can make a parallelogram fimilar to one given, and equal to the fum or difference of two rectilineal figures, though the method of finding their fum or difference has not been particularly pointed out. These are defects which ought undoubtedly to be removed; efpecially as these two problems are the moft ufeful of all in the Elements. It is therefore thought proper to alter the enunciations, fo as to preferve the fame meaning, and to conftru&t the 28th and 29th more fully, in confequence of which the demonftrations are a little fhortened. In the 28th, the parallelogram AP is faid by the ancients, to be applied to AB, deficient by the parallelogram PB, which is evidently fimilar to the given parallelogram EF. And in the 28th, the parallelogram AX is faid to be applied to AB, exceeding by the parallelogram BX, which is fimilar to the given one EL.

The cafe of Prop. D, which was omitted in the former editions, is now inferted. And Prop. E is added on account of its great ufe in geometry.

Book XI.

I'

BOOK XI.

DEF. VI.

Tis impoffible to demonftrate the properties and relations of folids contained by plane figures, without taking into confideration the angles which thefe planes make with one another; because their magnitude depends upon the angles made by their planes, as well as upon the magnitude of their plane figures. But in the propofitions concerning folids, in this book, there is no mention made of thefe angles. In the 25th, parallelopipeds are faid to be equal, when they are contained by equal and fimilar parallelograms: and in the 28th the prifms are faid to be equal, becaufe they are contained by the fame number of equal and fimilar planes. But folids may be contained by the fame number of equal and fimilar plane figures, and not be equal to one another, as fhall be made manifeft in the notes on the 10th definition,

definition, from Dr Simfon: It is alfo neceffary to their equali- Book XI. ty, that their equal and fimilar planes make equal angles with one another and the acknowledged accuracy of Euclid obliges us to believe, that this laft condition was not overlooked by him. And the 6th and 7th definitions must have been inferted for this very purpose, for they are of no other ufe in this book. But in the 6th definition, the inclination of two planes is faid to be an acute angle, whereas these angles may be right or obtufe angles; and therefore the word inclination, according to this defintion, is too limited to denote these angles. In the cafe of obtufe angles, we may produce one of the fides of the obtufe angle, and thus obtain an acute angle on the other fide of one of the planes but this would oblige us to introduce a condition into the propofitions where the term is used, which is not otherwife neceffary; and would very much embarrass the demonstrations as will be evident, if we confider what would be the cafe in plane figures, if by the word angle were meant only an acute angle. Thus, in the fourth propofition of the first book, not only muft the angles contained by the equal fides be equal, but thefe equal acute angles must be either both within the triangles, or elfe both without them, otherwife the propofition is not true; and the fame or greater embarraffments must be introduced into the other propofitions in which angles are concerned; and therefore the term angle, cannot be ufed in this limited fenfe; but either its meaning must be extended, or a new term invented. For the fame reason, we cannot use the word inclination in the limited sense given to it in this definition: nor can it be easily believed, that Euclid defined the inclination of planes, by an acute angle, for he ufes the word as in as extenfive a meaning as the word ywvice, in the 8th definition of the first book; and there is the fame neceffity for doing fo in this definition. It is therefore certain, that the word acute ought not to be in this definition; and as it might very readily have been inferted in it, from the preceding definition, through the inattention of fome tranfcriber, this is probably the way that it has come to be in it. It is therefore now thrown out, by which means the term inclination is rendered as extenfively useful in folids, as the term angle is in plane figures.

DE F. IX.

THIS definition has fewer conditions than what are neceffary, for folids may be contained by the fame number of fimilar planes, and not be fimilar, as Dr Simfon obferves. The fimilarity of plane figures is defined from the proportionality of their fides, and the equality of the angles contained by them; and in the

fame

BOOK XI. fame manner, the fimilarity of folids fhould be defined from the fimilarity of their containing planes, and the equality of the angles made by thefe planes. To this laft condition we are naturally led by the order of the definitions; for Euclid must have inferted the 6th and 7th definitions, för the very purpofe of determining the fimilarity of folids, because they can be of no other use in the doctrine of folids, and he certainly would not have given them without fome defign of using them. Dr Simfon thought that the condition wanting was the equality of their folid angles, and therefore he changed the order of the de finitions: but we have not fufficient reafon for thinking this; and the equality of folid angles cannot be proved, without going far away from the text of Euclid, unless the plane angles containing them be disposed in the fame order and fituation; which is a condition not neceffary to the fimilarity of folids. Befides, the equality of the folid angles is of no ufe in this matter, bût for determining the like inclinations of the planes, which can be better done without it, as shall be shown in the notes on Prop. A, B, and C, and the 26th propofition of this book. The order of the definitions and the condition wanting are now reftored.

DE F. X.

THIS definition is, that "Similar and equal folids are thofe that are contained by the fame number of fimilar and equal planes." Now this is not a definition, but a theorem, and it is not always true, as Dr Simfon has fhown by the following

example.

"Let there be any plane rectilineal figure, as the triangle a 12. 11. ABC, and from a point D within it, draw a DE at right angles to the plane ABC; take DE, DF equal to one another, upon oppofite fides of the plane; and let G be any point in EF; join DA, DB, DC; EA, EB, EC; FA, FB, FC; GA, GB, GC: Because the ftraight line EDF is at right angles to the plane ABC, it makes right angles with DA, DB, DC, which it meets in that plane; and in the triangles EDB, FDB, ED and DB are equal to FD and DB, each to each, and they contain b 4. 1. right angles; therefore the bafe EB is equal to the bafe FB: In the fame manner, EA is equal to FA, and EC to FC: and in the triangles EBA, FBA, EB, BA are equal to FB, BA, and the bafe EA is equal to FA; wherefore the angle EBA is equal to FBA, and the triangle EBA equal to the triangle FBA, and the other angles equal to the other angles; therefore these d 4.6.& 1. triangles are fimilar . In the fame manner, the triangle EBC Def. 6. is fimilar to FBC, and EAC to FAC; therefore there are two

c 8. I.