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BOOK I. the moderns are divided in their opinions about it. The ancients w did not admit experience to be judge of obvioufnels in geome

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try: But when they were obliged to affume any thing, they first made it as probable as they could, by demonftrating its converse, and other fimilar properties; after which they affumed it at once and this method is better adapted to geometry than the other; for thus the author's scheme is open to the view of his reader, who can then judge for himself of the propriety of the affumption. But if an author affume any thing from his own experience, which is not made probable from any thing he has faid before; he obliges his reader to take it entirely upon his word.

From this it follows, that the proper place of the 12th axiom is between the 28th and 29th propofitions, for it is not made fufficiently probable before the 28th. And much of its obfcurity would be removed by placing it there; for neither it, nor any that the moderns have substituted instead of it, have the fmalleft title to be ranked with the axioms.

The axioms which the moderns have preferred to it, may be reduced to three, though they may be expreffed in more ways

than one.

1. That a ftraight line perpendicular to one of two parallels, is also perpendicular to the other. Or it may be expreffed thus: If a perpendicular to a ftraight line drawn from a point of another, be not perpendicular to that other, the two lines will meet. These two are evidently of the fame order; for if parallels have a common perpendicular, ftraight lines which have not a common perpendicular, are not parallel, that is, they meet; and if ftraight lines which have not a common perpendicular do meet, it is evident, that those which do not meet, or are parallel, muft have a common perpendicular. This axiom was used by several authors of the last century: It may be demonftrated by the fifth of the preceding propofitions; and the 29th may be demonftrated from it nearly as in the fixth of this.

2. Perpendiculars to one of two parallels from points of the other, are equal to one another. Or, if two perpendiculars to a ftraight line from points of another be unequal, these lines will meet. Thefe may be fhewn to be of the fame order as the former were: They may be proved by the 7th of the preceding; and the 29th may be proved from them, by firft proving the former axiom, and then the 29th as in the fixth of the preceding.

3. A ftraight line which meets one of two parallels, meets alfo the other. Or, two ftraight lines cannot be drawn through the fame point, parallel to the fame ftraight line. One of thefe can be easily deduced from the other, as in the firft: They may

be

be proved by the help of the 7th preceding; and then the 29th Book I. may be proved from them: or rather may be omitted altogether; for by confining the parallels through the fame points to two, they determine the properties demonftrated in the 27th and 28th, to be the only effential properties of parallels.

We might have mentioned Dr Simfon's axiom in the beginning of his demonftration of the 12th axiom, which is certainly much more evident than any of the former, but is not fo eafily applied to the demonftration of parallels.

Of all these axioms, the first is preferable, because it can be demonftrated without the others, and the 29th may be proved from it with the greateft cafe. The fecond can only be used to demonftrate the firft; and the third cannot be proved, without first proving the firt or fecond, or elfe the 12th axiom. The first is indeed a particular cafe of the 29th, or of the 12th axiom; but, as was obferved, it is better to affume a particular cafe, than the general propofition, and its truth is much more probable : It is likewife inferior to the fecond, if experience be the proper judge of obvioufnefs; but this is counterbalanced by the eafe with which parallels are demonftrated by it; for they would thus be made as fimple as it is poffible to make them.

Of the 35th a more general demonftration is now given, by ufing a method of fubtraction, which was introduced into this propofition, by Dr Simfon, from M. Clairault's Geometry; and is one of the moft general and ufeful applications of the third

axiom.

The constructions of the 37th and 38th are a little altered, to avoid proving that the ftraight lines drawn parallel to the fides of the triangles muft meet the line joining their vertices, which was neceffary, according to the former constructions.

In the 39th, it is now proved, by the 12th axiom, that the ftraight line drawn through the vertex of one of the triangles, parallel to the bafe, muft meet a fide of the other triangle, as was neceffary here, because this is the first propofition in which it is required that the lines fhould meet. In the Greek, the demonftration is omitted the firft fix times, viz. in the 37th, 38th, 39th, 40th, 42d, and 44th; but is given fully in the feventh time, in the fame 44th, after which it is omitted throughout the Elements, except in the 10th of the fecond book and the 4th of the fixth, where it is as obvious as in any of the numerous places in which it is omitted. Simfon takes notice of the omiffion in one place, and concludes from it, that the demonftration has been spoiled, and the fame may be faid of the reft: To remove this irregularity, without embarraffing the demonftrations, a corollary is added to

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Book I. this propofition, on which the meeting of a line with parallels depends more immediately than on the 12th axiom.

In the conftruction of the 45th, it was required, that the parallelogram GM have an angle equal to E, from which it is proved, that KH is in the fame ftraight line with HM: But this is easier done by producing KH to M; and then it is only neceffary that the parallelogram GM have an angle equal to GHM, or have GHM for one of its angles.

BOOK II.

THE

BOOK II.

HERE are corollaries added to the 4th, the 5th, the 7th, the 8th, and the 10th propofitions, which are chiefly intended to exprefs the enunciations of thefe propofitions in a more general way, as is done by the moderns.

The demonftration of the 8th is made fhorter, by proving BK to be equal to KR, and BN together with PL to be equal to each of the rectangles AK, MR, KF.

The conftruction of the roth is now made fimilar to that of the 9th, by which means they may both be demonstrated in the fame words, and therefore it was needlefs to repeat the demonftration. In the Greek, it is firft affumed, that DF meets a parallel to AD; and then it is demonftrated, that EB meets a parallel to EC. They are cafes that have often occurred, and therefore their demonftration might have been omitted, fince it ought to have been given before; but if either of them be demonstrated, it ought undoubtedly to have been the first mentioned.

Dr Simfon has divided the 13th into three cafes; but the two firft can be demonftrated in the fame words: and the demonstration of the third is unneceffary, because it is manifeft from the 47th of the first book.

There are three propofitions added to this book, because they are very often used by geometers, and are connected with this part of the subject.

BOOK

BOOK III.

DEFINITION S.

HE firft definition is left out, because it is not a definition, Book III.

TH

but a theorem, as Dr Simfon has shown.

The 7th, which is the angle of a fegment, is left out, be

cause it can be of no use, and has done much mischief.

And the definitions of an arch, and of a chord, are inserted, because they are continually used by geometers..

PROPOSITIONS.

IN the 2d, the fuppofition by which the demonftration was made indirect, is now left out; and it is then direct and complete; for it follows immediately from the definition of a circle, that a point between the centre and the circumference is within the circle; and this is directly proved in the demonftration, when it is shown, that DE is lefs than DF, or that E is between D and F. And it is certainly as evident, that the point E in the radius DF is within the circle, as it is that a point without the circle is in the radius produced, which is affumed in the indirect demonstration.

The 5th and 6th have the fame construction and demonstration. Befides, it is affumed in the 6th, that two circles can touch one another, which is not evident, until the 7th has been demonftrated. The poffible existence of things is not to be affumed, any more than their properties. Euclid affumes the existence of a plane, a straight line, and a circle; but before he supposes any thing else to be poffible, he either exhibits it in a construction, or makes it evident by an example, which is generally given in the moft fimple cafe. Thus, he gives an example of a triangle, and confequently of an angle in the first propofition, and of a perpendicular in the 11th, and of parallels in the 27th, and of a parallelogram in the 33d, and of a square, and confequently of a rectangle, in the 46th, and this always before he affume them in an hypothefis. This is the general rule which regulates the arrangement of the whole Elements, and it is in a great measure to it that they are indebted for their fuperior beauty and elegance: and wherever it is departed from, it is owing either to the infertion of a propofition which is not Euclid's, or to a change in his arrangement: Which ever of these be the cafe with this propofition, and the fourth of the fifth book, which are the only places

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Book III. in which the rule is tranfgreffed, it is proper to remove the blemifh and this is now done here by putting the word "meet," inftead of "cut," into the enunciation of the 5th, and by throwing out the 6th, because it is then included in the 5th.

There feems to have been a series of alterations introduced into the Elements, from the 9th to the 13th, by fome editor, who intended either to change Euclid's arrangement, or to exclude the 7th and 8th from the Elements, on account of their length; and who did not perceive that the hypothefis of the 11th, 12th, and 13th, depended on the 7th and 8th, as well as their demonftrations. The first demonftrations of the 9th and 10th, are generally allowed not to be Euclid's, for they depend upon a principle which he no where makes ufe of; and he could have demonstrated them eafier, even without the 7th and 8th, by the help of the 24th of the first book: and if the author of these two demonftrations gave also a demonftration of the 11th, we may conclude, that he would have placed it before that of Euclid: But the fecond demonftration of the 11th is very much vitiated, for part of it is omitted, and part of the first demonftration is tranfcribed into it. This, however, is probably owing to the careleffnefs of tranfcribers, who have made the 9th to depend upon the first part of the 7th, instead of the last part of it, and left out the word "circumference" in the 10th: and the fhortnefs of the firft demonftration of the 11th, might have made the corruptions of the fecond to be lefs attended to. This fecond demonftration has two cafes; one when the centre of the greater circle is within the lefs, and the other when it is without it: and this diftinétion of cafes can be of no ufe, unless the demonstration depend upon the 7th or 8th: confequently, we are led by this claufe to the following demonftration. First, Let F be within the circle ADE: and becaufe F is a point in the diameter of the circle ADE, which is not the centre, FD which paffes through the centre is greater than FA: but FH is equal to FA; therefore FD is greater than FH; which is abfurd. In like manner, it may be proved, by Prop. 8. that the fame abfurdity would follow, if the point F be without the circle ADE. Therefore, &c. This appears to be Euclid's demonftration; and if we use the point G, instead of F, to avoid the distinction of cafes, it is the most natural, eafy, and elegant demonftration, that can well be conceived. And the 12th can be demonstrated in the fame way. All these corrections are now made: and the fecond cafe of the 13th is rejected, because Dr Simfon's demonftration of the firft cafe is general, whether the circles touch on the infide or the outfide.

A direct demonftration is given of the 16th, for the rea fons given in the notes on the fecond. It is evident, that

the

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