It is to be hoped what has been faid will be fufficient to induce the reader to pay a proper attention to the structure of the demonftrations made use of in this book, which have a wonderful elegancy and force even when taken fingly; but more particularly when the arrangement of the whole is confidered: for the better understanding of which I would advise the reader to divide the book into four portions. The nature of ratios fhews the propriety and neceffity of introducing the properties of equimultiples first of all; and although the fourth propofition is not absolutely confined to its place; yet as it depends upon the third, it would be difficult to give it a more advantageous pofition. The first fix propofitions therefore will make the first portion. Again, we may have a very distinct notion of this ratio of magnitudes without knowing what the magnitudes themselves are directly: Here therefore fome rule or teft is neceffary for enabling us to get at the common relations of greater, equal and lefs, when we have only this ratio to guide us. The ftudent is therefore to confider the feventh, eighth, ninth and tenth propofitions, as making a diftinct section, and as introduced for the very purpose of fettling this point.

It might appear to a fuperficial obferver, who had just come from proving that if four magnitudes are proportionals, they will be proportionals by inverfion; that a fimilar method of reasoning might be applied to fhew that they will be proportionals by alternation. But it happens here, that the equimultiples, of the firft and third, of the magnitudes which we want to prove to be proportionals; are not equimultiples of the second and fourth of those magnitudes which are proportionals by the fuppofition; but of the first and second; which deftroys all profpect of fucceeding according to that plan. And indeed this demonstration is no easy matter, for all the propofitions from the tenth to the feventeenth may be confidered as introduced to demonstrate; that magnitudes will be proportionals by alternation. And this will make up the third portion according to the divifion above mentioned. And the remaining part of the book may be confidered under one head, in which is proved that magnitudes will be proportionals; by compofition divifion, converfion, and by the two methods of reasoning by equal distances. And thus I think a very distinct and comprehenfive view

of the fubject of this book may be taken, and fuch an one as feems likely to fix it in the memory.

AGREEABLE to the rule already laid down, instead of remarks upon particular demonftrations, I fhall reduce what I have to say under distinct heads, which the reader may apply as he goes along; but first it seems neceffary to be a little particular, as to the construction of the thirteenth propofition. Indeed there is nothing which feems to be fo prevailing an error, with regard to this book, as a neglect of the constructions. And yet this is a thing which Euclid is, every where, more particularly attentive to. He fometimes leaves his reader to draw the consequence from a construction, when it is fufficiently obvious, and cannot be done shortly and elegantly; an inftance of which we have in the first propofition of the third book; where there are two conftructions, but he reasons directly, only from one of them, and then joins both their confequences in his conclufion: for the demonftration only proves that the center must be in the perpendicular; for no abfurdity will follow from the reasoning, by supposing the point G to be any where in the line CD; not even if we fuppofe it to be at the circumference; for it will not then be abfurd to fuppofe the angle GDB equal to CDB; we have therefore Euclid's authority for drawing consequences in this manner; and I have not the least doubt of this demonftration being the genuine one given by Euclid; because to have made it quite regular the two conftructions must have been feparated; and then it would not have been one, but two diftinct propofitions; in the first of which it was to be proved that the center must be in the perpendicular; and in the fecond, that this perpendicular cut in halves would find the center; which is a much greater formality than the nature of the problem requires, although the ftudent ought to examine it in this manner. But in this thirteenth propofition the cafe feems to be very different, and confequences are drawn from a construction which I

believe

believe it will be found very difficult to perform. Simfon in his note upon this proposition says; "In Commandine's, Brigg's and Gregory's tranflations, at the beginning of this demonstration, "it is faid, and the multiple of C is greater than the multiple of D; " but the multiple of E is not greater than the multiple of F; which “words are a literal translation from the greek: But the sense evi"dently requires that it be read, fo that the multiple of C be greater than the multiple of D; but the multiple of E be not "greater than the multiple of F." Now I am so much of a contrary opinion, that it appears to me that this change destroys every veftige of the true conftruction of this propofition. My notion is that the proper equimultiples are given me by the hypothefis; and, the multiples and magnitudes being given, I can measure the one by the other; and confequently know how many times the magnitudes are taken; from which the conftruction is obvious but if I have nothing to direct me but the magnitudes themselves; concerning which also there is no fuppofition of greater or less, I should be glad to know in fuch circumstances, how this conftruction is to be performed; I know it is faid let them be taken, which if I were to add my conjectures after the manner of fome commentators, I would fufpect to be the correction of fome hafty editor who did not understand the demonstration..

In proving four magnitudes to have the fame ratio, the firft to the fecond which the third has to the fourth; it is neceffary that the equimultiples of the first and third be any whatever; and also, that the equimultiples of the fecond and fourth be taken at a venture. Now it is obvious that if they be taken as often as one de-. terminate magnitude contains another, they cannot be any whatever; neither is the fum of two multiples, each of which has been. taken at a venture, to be confidered as any multiple whatever; because if in one case it should happen to be taken three times, and. in the other five times; these multiples added together, would not happen to be eight times that magnitude, but muft really be eight times the magnitude, not accidentally but confequentially: again when a multiple of a magnitude is taken, and then a multiple of that multiple as in the third propofition; this laft is not any multiple whatever of the first magnitude; for inftance the first happens,

to

to be taken three times, and this multiple again five times; now the last multiple does not happen to be fifteen times the firft, magnitude, but must be really fo. And this will be fufficient to direct the student to the proper ufe of the phrase any whatever.

;

Simfon's remarks upon the eighteenth propofition are very ingenious; for the poffibility of the fourth proportional is not to be taken for granted: because this is a very different notion from that which fuperficial people are apt to confound with it: for when two magnitudes are made the fubject of contemplation, their exiftence is a part of the fuppofition; and if they do exift, it must be under one or other of these forms; A must be equal to B, or greater, or lefs; and if I prove that A cannot be greater than B nor less, I certainly demonftrate that A is equal to B. But it is very different where the existence of the magnitude may be called in question, which is very reasonably done in the prefent inftance; for it may be said to Euclid you have confined yourself so much by your definition of the fame ratio, that a fourth proportional may be really impoffible according to your definition but when you have once shewn the poffibility of it, I will then admit your reafoning. It is very certain, that we have fufficient principles, before this propofition, for finding a fourth proportional to any three given straight lines, which would make the demonstration unexceptionable as we have it at prefent; by proceeding thus; to AB, BE and CD find a fourth proportional, which must be greater, equal or less than FD &c; but this would be mixing things together, which our author certainly intended to keep feparate. But to conclude this chapter, whoever has a proper notion of a part and multiple; and knows what is required to prove that four magnitudes have the fame ratio the firft to the second which the third has to the fourth, can never be mistaken in forming a judgement of the demonstrations contained in this book. Euclid conftantly fuppofes his magnitudes to be straight lines as is obvious from his conftructions and demonstrations; so that there is no occafion to fuppofe the magnitudes in any propofition to be of the fame kind; and indeed thofe critics who have made fuch alterations, if they had understood their business, ought at least to have changed the enunciations of half the propofitions in the book.

CHAP.

С НА Р. IX.

Of the fubject and arrangement of the fixth book.

OUR author having laid down this general doctrine of parts and multiples proceeds to apply it to the investigation of such properties of the triangle, rectangle and circle as could not be obtained by the former method of comparison. For what is delivered in the fifth book itself cannot, ftrictly speaking, be confidered as properties of triangles, parallelograms &c. but only as properties of parts and multiples of fuch magnitudes as come under the two first definitions. But in this book he treats of the triangle as fuch, and thews when triangles have the fame altitude, they have the fame ratio to one another, as their bafes. And here the thoughtless reader will do well to obferve; that every triangle may have its altitude exprest by three different straight lines, as each fide can be the base for the altitude of a triangle is a relative term, and has no meaning until the bafe is fixt. From this he demonftrates that if the two fides of a triangle be cut by a straight line parallel to the third fide, they will be cut in the fame proportion; this is the fundamental principle which runs through the whole of this book; and ought for this reason to be examined in every point of view. For after the reader has confidered this property according to its moft obvious acceptation, he will find it more general than he fufpected, by making it a property of two indefinite ftraight lines interfecting one another, and cut in any manner by two parallel lines.

It has been mentioned already that the triangle is the great inftrument of geometrical investigation, and Euclid's first object in this book is to lay down thofe principles upon which the fimilarity of triangles depends; and by this acquifition the power of the science becomes aftonishingly great, as it is difficult to fay what cannot be done by it that is really practicable; the judicious reader therefore will examine the firft eight propofitions with that attention which the importance of the subject requires. The five following propofitions teach how to divide a straight line into any number of equal parts; to cut a given line in the fame proportion,

as