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Book I. the moderns are divided in their opinions about it. The ancients mdid not admit experience to be judge of obviousness in geome

try: But when they were obliged to assume any thing, they first made it as probable as they could, by demonftrating its converse, and other similar properties ; after which they assumed 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 assumption. But if an author assume 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 sufficiently probable before the 28th. And much of its obfcú. rity would be removed by placing it there; for neither it, nor any that the moderns have substituted instead of it, have the smallest 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 expressed in more ways than one.

1. That a straight line perpendicular to one of two parallels, is also perpendicular to the other. Or it may be expressed thus : If a perpendicular to a straight 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, straight lines which have not a common perpendicular, are not parallel, that is, they meet; and if straight lines which have not a common perpendicular do meet, it is evident, that those which do not meet, or are parallel, must have a common perpendicular. This axiom was used by several authors of the last century: it may be demonstrated by the fifth of the preceding propositions ; and the 29th may be demosiftrated 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 straight line from points of another be unequal, these lines will meet. · These

may

be fewn to be of the same order as the form mer were : They may be proved by the 7th of the preceding; and the 29th may be proved from them, by first proving the former axiom, and then the 29th as in the fixth of the preceding

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

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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 same points to two,
they deterinine the properties demonstrated in the 27th and 28th,
to be the only effential properties of parallels.

We inight have mentioned Dr Simson's axiom in the begin-
ning of his demonítration of the 12th axiom, which is certainly
much more evident than any of the former ; but is not so easily
applied to the demonstration of parallels.

Of all these axioms, the firit is preferable, because it can be demonstrated without the others, and the 29tii inay be proved from it with the greatest ease. The second can only be used to demonstr

rate the firit ; and the third cannoi be proved, without first proving the first or second, or else the izth axiom. The firit is indeed a particular case of the 29th, or of the 12th axiom; but, as was observed, it is better to assume a particular case, than the general propofition, and its truth is much more probable : It is likewise interior to the second, if experience be the

proper judge of obviousness; but this is counterbalanced by the ease with which parallels are demonstrated by it; for they would thus be made as imple as it is possible to make them.

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Of the 35th a more general demonstration is now given, by using a method of subtraction, which was introduced into this propofition, by Dr Simfon, from M. Clairault's Geometry; and is one of the unoft general and useful applications of the third axioin.

The constructions of the 37th and 38th are a little altered, to avoid proving that the straight lines drawn parallel to the sides of the triangles must meet the line joining their vertices, which was necessary, according to the former constructions.

In the 39th, it is now proved, by the 12th axiom, that the straight line drawn through the vertes of one of the triangles, parallel to the base, must meet a side of the other triangle, as was necessary here, because this is the first proposition in which it is required that the lines should meet. in the Greek, the demonftration is omitted the first fix times, viz. in the 37th, 38th, 39th, 40th, 42d, and 44th; but is given fully in the seventh time, in the same 44th, after which it is omitted throughout the Elements, except in the 10th of the second book and the 4th of the fixth, where it is as obvious as in any of the numerous places in which it is omitted. Dr Simson takes notice of the omiffion in one place, and concludes from it, that the demonstration has been spoiled, and the same may be faid of the rest : To remove this irregularity, without embarrassing the demonstrations, a corollary is added to

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

In the construction 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 same straight line with HM: But
this is easier done by producing KH to M; and then it is only
necessary that the parallelogram GM have an angle equal to
GHM, or have GHM for one of its angles.

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Book II. THERE are corollaries added to the 4th, the sth, the 7th,

the 8th, and the roth propositions, which are chiefly intended to express the enunciations of these propositions in a more general way, as is done by the moderns.

The demonstration of the 8th is made shorter, hý proving BK to be equal to KR, and BN together with PL to be equal to each of the rectangles AK, MR, KF."

The construction of the joth is now made similar to that of the 9th, by which means they may both be demonstrated in the same words, and therefore it was needless to repeat the de. monstration. In the Greek, it is first affumed, that DF meets a parallel to AD; and then it is demonstrated, that EB meets a parallel to EC. They are cases that have often occurred, and therefore their demonstration might have been omitted, since it ought to have been given before ; but if either of them be demonstrated, it ought undoubtedly to have been the first mentioned.

Dr Simson has divided the 13th into three cases; but the two first can be demonstrated in the same words : and the demonstration of the third is unnecessary, because it is manifest 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.

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

The 7th, which is the angle of a segment, is left out, because 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.

PROPOSITION S.

In the 2d, the supposition by which the demonstration 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 less 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 assumed in the indirect demonstration.

The 5th and 6th have the same constru&tion and demonstration. Besides, it is assumed in the 6th, that two circles can touch one another, which is not evident, until the 7th has been demonstrated. The possible existence of things is not to be assumed, any more than their properties. Euclid assumes the existence of a plane, a straight line, and a circle ; but before he supposes any thing else to be possible, he either exhibits it in a construction, or makes it evident by an example, which is generally given in the most simple case. Thus, he gives an example of a triangle, and consequently of an angle in the first proposition, and of a perpendicular in the rith, 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 aisume them in an hypothesis. 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 superior beauty and elegance : and wherever it is departed from, it is owing either to the insertion of a propofition which is not Euclid's, or to a change in his arrangement: Which ever of these be the case with this propofition, and the fourth of the fifth book, which are the only places 002

in

Book III. in which the rule is transgressed, it is proper to remove the ble.

milh: and this is now done here by putting the word “meet,” instead of “ cut," into the enunciation of the 5th, and by throwing out the 6th, because it is then included in the sth.

There seems to have been a feries of alterations introduced into the Elements, from the oth 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 hypothesis of the 11th, I 2th, 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 use 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 demonstrations gave also a demonstration of the 11th, we may conclude, that he would have placed it before that of Euclid : But the second demonstration of the 11th is very much vitiated, for part of it is omitted, and part of the first demonftration is transcribed into it. This, however, is probably owing to the carelessness of transcribers, 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 thortness of the first demonstration of the 11th, might have made the corruptions of the second to be less attended to. This second demonstration has two cases; one when the centre of the greater circle is within the less, and the other when it is without it: and this distinction of cases can be of no use, unless the demonstration depend upon the 7th or 8th: consequently, we are led by this clause to the following demonstration. First

, Let F be within the circle ADE: and because F is a point in the diameter of the circle ADE, which is not the centre, FD which passes through the centre is greater than FA: but FH is equal

to FA ; therefore FD is greater than FH; which is absurd. In Jike inanner, it may be proved, by Prop. 8. that the same ab.

dity 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 cases, it is the most natural, easy, and elegant demonftration, that can well be conceived. And the 12th can be demonstrated in the same way.

All these corrections are now made : and the second case of the 13th is rejected, because Dr Simson's demon. Atration of the first case is general, whether the circles touch on the inside or the outside.

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

the

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