the fecond part is not Euclid's, but is put in by fome Book III. editor, probably the fame that introduced what we find in this and the 31st, about the angle of a fegment; because its only ufe is to fhew, that but one straight line can touch the circle in the fame point: and this is alfo the only ufe of the 18th and 19th; for perpendiculars being fully determined in the firft book, it cannot be fuppofed, if Euclid had proved in the 16th, that the perpendicular to the radius at its extremity is the only line that touches the circle there, that he would also have proved, in the 18th, that this radius is perpendicular to the tangent.

A more general demonftration is given of the 21ft, fimilar to what Dr Simfon gives of the fecond cafe, for that given in the Greek is applicable only to angles in a fegment greater than a

femicircle.

A very useful corollary is added to the 22d, and another to the 35th.

There are four propofitions added to this book, of which the two first and their corollaries contain the converses of the most useful propofitions in this book, which are as often used as the propofitions themselves: and the other two contain very useful properties, the firft of the circle, and the other of the triangle.

IT

T is now demonftrated by the 12th axiom, that the fides meet one another, so as to form the triangle LMN about the circle in the 3d propofition; and that the perpendiculars DF, EF in the 5th meet one another; and that the straight lines touching the circle in the 12th meet one another; because these cafes are not fimilar to that for which the corollary to the 39th of the first book was given.

It is likewife demonftrated in the third, that the point L in which LM, LN meet falls above A, C; for if they were to meet below MN, the triangle thus conftructed would not be described about the circle ABC, according to the fourth definition: Nor would it be equiangular to DEF.

BOOK

BOOK IV.

BOOK V.

THE

BOOK V.

DEFINITION S.

THE 3d and 8th are rejected, because they are unneceffary and useless; and the 9th is included in definition D.

And there are fix new definitions introduced; the first of which relates to the prefent method of expreffing the 5th and and 7th definitions: but the reft ought to have been in the Elements before; and the want of them is one of the causes of the obfcurity in which the fubject of proportion has been hitherto involved.

In order to understand the 5th and 7th definitions, it is neceffary to confider in what way we acquire our ideas of ratios.

Some geometers feem to think, that our ideas of the proportion of magnitudes are derived from our ideas of the nature of abstract numbers. But we certainly acquire our first ideas of proportion from external objects, in the fame manner that we acquire our ideas of numbers. By obferving one magnitude to be double of another, we acquire the idea of a particular ratio, or relation, that the greater has to the lefs: and when we afterwards find two magnitudes, one of which is alfo double of the other, we say, that they have the fame ratio which the two former have to one another, or that the four are proportionals. In like manner, by obferving one magnitude to be triple, quadruple, or any multiple of another, we acquire ideas of other ratios and by proceeding in this way, we obtain ideas of all the ratios belonging to the first clafs of commenfurable magnitudes; that is, when the greater is a multiple of the lefs. So that all these ratios are obtained by conceiving the leffer magnitudes to be added to themselves fome number of times. Nor is it more difficult to conceive how we may obtain ideas of all the ratios belonging to the fecond clafs of commensurable magnitudes; that is, thofe of which the greater is not itself a multiple of the lefs, but of which fome multiple of the greater is also a multiple of the lefs. It is only by conceiving the greater to be added to itself continually, until the multiple contain the less exactly; and then the ratio of the greater to the lefs is obtained. For the ratio of A to B is determined, by faying, that four times A is equal to feven times B, as properly as by saying, that A is greater than B, by three fourths of B. In this manner, we may acquire ideas of the ratios of all magnitudes which have common multiples. The method may appear to be tedious, but it is fimple and obvious.

Hence

Hence it appears, that a ratio is determined when a certain Book V. multiple of the greater is found to be a multiple of the less; and that this ratio is diftinguished from every other ratio, by the magnitudes to which it belongs, having these multiples equal; and therefore we conclude, that the ratio of two magnitudes is the fame with the ratio of two other magnitudes, when fome equimultiples of their antecedents are alfo equimultiples of their confequents: But that if one of the equimultiples of the antecedents be a multiple of its confequent, and the other not the fame multiple of its confequent; in that cafe, the ratios are not the fame.

Again, whenever we find that there are fome equimultiples of the antecedents, fuch that one of them is lefs than a multiple of its confequent, but the other not less than the fame multiple of its confequent; we need not inquire for equal multiples of the antecedents and confequents, but may conclude, that though we should find equimultiples of the antecedents, fuch that one of them is a multiple of its confequent, the other would not be the fame multiple of its confequent.

Let twice A be less than three times B, but twice C not leís than three times D; then, although we fhould find that three times A is equal to four times B, three times C is not equal to four times D. For twice C not being less than three times D; by tripling them, fix times C is not lefs than nine times D; it is therefore greater than eight times D; and taking their halves, three times C is greater than four times D. In the fame manner, if four times C were found to be equal to seven times D, it may be proved, that four times A is less than seven times B.

On the contrary, if three times A be equal to four times B, but three times С not equal to four times D; equimultiples of A and C can be found, fuch that one of them is lefs, and the other not lefs, than equimultiples of B and D. If three times C be less than four times D, this is evident, for three times A is not less than four times B. But let three times C be greater than four times D by the magnitude E; then fome number of times E fhall exceed C; let this be three times E; and because three times C is equal to four times D, together with E; by tripling them, nine times C is equal to twelve times D, together with three times E: and three times E is greater than C; therefore eight times C is greater than twelve times D: and three times A being equal to four times B; by tripling them, nine times A is equal to twelve times B; therefore eight times A is less than twelve times B; and eight times C is not lefs than twelve times D: Wherefore eight times A and eight times Care equimultiples of A and C, fuch that one of them is lefs,

and

Book V. and the other not lefs, than twelve times B and twelve times D, which are equimultiples of B and D.

Hence it is manifeft, that if there be not fome equimultiples of the antecedents, fuch that one of them is lefs than a multiple of its confequent; and the other not less than the fame multiple of its confequent, the magnitudes are fo related to one another, that if one of the equimultiples of the antecedents be a multiple of its confequent, the other is the fame multiple of its confequent and therefore the magnitudes have the fame ratio.

There is therefore only one cafe in which the ratios are not the fame, viz. when one of the equimultiples of the antecedents is lefs than a multiple of its confequent, and the other not lefs than the fame multiple of its confequent: and in this cafe, we fay, that one of them is greater than the other; that is, we fay, that the antecedent of which the multiple is not less than that of its confequent, has a greater ratio to its consequent than the other antecedent has to its confequent: Or, because the multiple of the former antecedent contains its consequent oftener than the fame multiple of the other antecedent contains its confequent; we fay, that the first ratio is greater than the other, when fome multiple of the first antecedent contains its confequent oftener than the fame multiple of the other antecedent contains its confequent.

And because this is the only cafe in which the ratios are not the fame, we conclude, that two ratios are the fame, when taking any equimultiples whatsoever of the antecedents, they are either both lefs, or else both not lefs, than any equimultiples of their confequents: Or, because in this cafe the equimultiples of the antecedents contain their confequents the fame number of times, we fay, that two ratios are the fame, when all the equimultiples of the antecedents contain their confequents equally.

This is a more fimple expreffion of the definition of equal ratios, than that given by Dr Simfon; for the literal tranflation from the Greek, though shorter than his, is very obfcure: But the meaning is very nearly the fame. In the Greek, the equimultiples of the antecedents are required to be both lefs, or both equal, or both greater, than the equimultiples of the confequents; but here they are required to be both lefs, or both not lefs. It was, however, fhewn before, that unless one of them be lefs, and the other not lefs, it will always happen, that if one of them be a multiple of its confequent, the other is the fame multiple of its confequent; and therefore the definitions are the fame: the generality of the multiples making all particular differences to vanish. For it is to be obferved, that the multiples taken, either according to the definition now given,

or

or according to that in the Greek, must be general, representing Book V. any multiples whatsoever.

But, befides fimplicity of expreffion, the definition now given has the advantage of the other, in its being, in moft cafes, more eafily applied to the purpofe of demonftration, as will be mani feft to any one who chufes to compare the demonstrations of the 17th and 18th, or even of those from the 7th to the 13th in this edition, with thofe in the other editions. The demonftrations depend on the fame principles, and are conducted in the fame manner, with those in the former editions; but the fimplicity of the constructions often renders fewer fteps neceffary in the de monstrations.

PROPOSITIONS.

THE I, 2d, and 6th, are made more general than they are in Euclid. They belong to all magnitudes which contain others equally, as well as to equimultiples of them.

That beginners have the 4th for the first propofition that they read about proportionals, is one of the caufes why they find this book fo difficult to be understood by them. The words multiple and equimultiples are new to them, and it is with fome difficulty that they can bring themfelves to believe, that these ftrange words are expreffions of ideas with which they are familiarly acquainted. But they begin to get over this difficulty, and to be reconciled to thefe words, when they fee the ufe that is made of them in the axioms, and the three first propofitions. In this propofition, however, a new difficulty arifes from the ufe that is made in it of the fifth definition. In order to exprefs that definition intelligibly in the English language, a good deal of circumlocution is neceffary; and this renders it difficult for the reader, to connect the feveral parts of it, and to form a fingle, complete, and juft idea of the whole: and this difficulty is increased, rather than removed, by the abrupt way in which it is introduced into this demonftration. Whatever idea the reader may have formed of the 5th definition itself, he must be allowed to be altogether unacquainted with the method of applying it to the purpofe of demonftration, for he has not yet had an opportunity of feeing it ufed that way. He must also be allowed to doubt the existence of fuch magnitudes as have all the properties mentioned in that definition, for this ought not to be taken for a poftulate; and there are abundance of examples by which their poffibility may be fhewn, one of which is given in the 7th, and another in Prop. C. But they are not given before this propofition; and until they be given, the mind must be allowed to doubt. Befides, the reader's mind is pre

PP

occupied