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Boox I. and second poftulates, for we certainly understand by these poo nftulates, that only one straight line can be drawn between the

fame two points, and that a terminated ftraight line can be produced only in one direction; fo that the position of the points determines the pofition of the line ; it is therefore probable, that Euclid understood it as included in his tenth axiom; for that it was not his definition, is evident from what has been said, and because it is his 26th proposition of the Data, which is the 29th of Simfon's edition. Besides, it gives too remote an intimation of the nature of ftraight lines, and their properties cannot be tafily deduced from it.

The other definition is, that a straight line is that which canAot cut another straight line in more points than one. In the Greek, this is given in the first proposition of the eleventh book, as the reason why two straight lines cannot have a common fegment; and this latt is affumed every where in the Elements. Their cutting only in one point seems to be a particular cafe, or at least a consequence of that facility of coinciding which, as was observed, Euclid makes the criterion of straight lines; it may therefore very properly be made an axiom: and it feems neceffary to do this, becaule, when we are not actually applying straight lines to one another, it may be found that two points of one of them coincide with two points of anotber, in which case the coinciding of the lines follows more immediately from the axiom, than from their uniformity: and it is probable, that Euclid understood his tenth axiom in all this extent: But to make it a definition, would be attended with disadvantages, for then the only criterion of the coinciding of straight lines, would be their meeting in two points : To suppose that they coincide, without first finding that they meet in two points, would be to aflume a property of thein, not contained in the definition: We are not therefore warranted to say, as in the fourth propofition, Let the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon DE ; for all that follows from AB and CD being Atraight lines, ascording to this definition, is, that they would coincide, if, besides A being on D, some other point of AB be thown to coincide with some other point of DE. Farther, the property contained in this definition is not the same with the principle upon which the mechanical description is founded, as might easily be made manifeft; not to mention its being negative, and not derived from inherent properties.

In like manner, no geometrical definition of an angle has been given ; and accordingly, among the ancients, different definitions were given by different authors. Thus, Euclid called it adipis, che inclination of lines: Apollonius called it cuyaywyn, the draw

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ing together of a superficies to a point; and others called it the Book I.
first distance of the containing lines: And the philosophers dif-
puted whether it was a quantity, or a quality, or a relation, or
whether it belonged to one or to several of the categories. The
moderns hare generally followed Euclid, or have called it the
aperture or opening of two lines that meet. But no author has
ever deduced the properties of angles from his definition; nor is
it easy to do so without metaphysical reasoning. Thus we find,
that nothing in the Elements depends upon these nine definitions;
and therefore the purpofe of inftruction would be better obtained
by removing them froin the definitions, and explaining them in
an introduction.

Besides these, there are several other definitions which might
have been omitted, or altered for the better. Thus, in the defi-
nition of an equilateral triangle, there is nothing peculiar to a
triangle, for every rectilineal figure is in the Elements said to
be equilateral, when it has all its fides equal. And the acute
and fcalene triangles are never mentioned in the Elements, nor
have they any peculiar properties : They may indeed be said to
be necessary for completing the arrangement of triangles; but
even this cannot be pleaded for the definitions of quadrilaterals,
for the classification of them in the Elements is complete, and
altogether different from that in the definitions; fo that the de-
finitions are not well adapted to the work itself, and might have
been altered with advantage.

It shall only be farther remarked, that the whole design of definitions is, to give names to things that have certain properties; and that unless the properties ascribed to them be their generation, the definitions give no foundation from which to conclude the existence of the things defined, as Dr Barrow obferves. It is not therefore regular to deduce corollaries immediately from definitions, because corollaries, as well as other propositions, suppose the existence of the things concerned in them. Accordingly, Euclid either assumes their existence in his poftu. lates, or exhibits it in a conftruction, or deduces it from what has been previously established, before he deduce any of their properties from their definitions.

A X I O M S.

The only alteration that is made here, is in the 10th axioma,

which is now made more extensive than it was before.

It includes not only the axiom, That two straight lines cannot inclose a space, but also the other, That two straight lines cannot have a common segment: This last is every where supposed in

the

Book I. the Elements, though it be no where repeated, except in the firft

propofition of the eleventh book.

For example, it is assumed in the second poftulate; for, as Proclus observes, that postulate would be nugatory, if the terminated straight line could be the common segment of two Itraight lines. In like manner, as Zeno objects, the firkt proposition is not sufficiently demonstrated, without affuming, that two straight lines cannot have a common segment: For if the straight lines AC, CB can have a common segment, the sides of the triangle may be only parts of them, between its extremity and the points A and B; and therefore they may be less than AB; so that the triangle may not be equilateral. Likewise, without this supposition, the fourth would not be sufficiently demonstrated; for when AB is applied to DE, if they could have a common segment less than AB, they would coincide the length of this segment; but not afterwards; so that the point B might not coincide with the point E.

The fifth depends on the same supposition ; for if either side of the isosceles iriangle, as AB, be the common segment of two ftraight lines, these lines would make unequal angles with the base BC on the other side of it: But each of them would be equal to the angle BCE, by the fifth prop. In the same manner, it may be shown, that the same thing is supposed in many other propofitions ; and therefore it is evident that Euclid confidered it as an axiom.

As to ašioms, it is generally allowed, that they ought to be self-evident, and likewile as few as possible ; for nothing ought to be considered as self-evident that is capable of demonstration.

The postulates ought likewise to be as few as poslīble ; for, as Sir Isaac Newton obseryes, poftulates are principles which geometry borrows from the arts, and its excellence consists in the, paucity of them. The postulates of Euclid are all problems derived from the mechanics.; but there are things assumed by geometers, which are neither self-evident nor problems; such as that called the 12th axiom: and it is plain, that such assumptions ought not to be made, but when the thing affumed is demonstrated elsewhere, or when it cannot possibly be demonstrated ; for all certainty arises from self-evidence. Poftulates likewise limit the geometer in his definitions, for that property of a line or figure on which its mechanical description is founded, is the only property from which the assumer must define it; otherwise his constructions would not agree with his demonftrations, Thus Euclid could not define his circle from any property but that of its having equal radii, because it is described with the same opening of the compasses. And because a straight line is drawn either by ftretching a thread, or by applying another

straight

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Atraight line to the place where it is to be drawn, Euclid's cha- Book I.
racter of straight lines must be the facility of applying them to
one another, or the property of a stretched thread, which is its
invariably affuming the same position, as often as it is stretched
in the same situation.

Many of the moderns complain, that Euclid, by taking so
little for granted, las narrowed the foundation of geometry too
much. And they maintain, that we are at liberty to assume any
thing, the existence of which does not imply a contradi&ion, or
involve an impossibility. But were this the case, we might af.
sume every thing, for we certainly suppose nothing impossible,
when we affume, that fimilar rectilineal figures are to one another
in the duplicate ratio of their homologous fides; for Euclid has
proved this to be certainly true. They perhaps mean things
evidently possible ; but many things may appear evident to a
writer, which are not so to a learner. An author of Elements.
ought to confider the fate of his reader's mind, who, being alto-
gether unacquainted with geometry, can admit nothing to be
felf-evident that is peculiar to it. It is necessity, and not liberty;
which is the cause of assumptions; and the writer who assumes
what he is not obliged to assume, is not writing for his reader's
instruction, but is endeavouring to beg fame, by humouring his
reader's indolence.

It follows from this, that affumptions ought to be as particular as possible; for example, we ought not to assume a general proposition, if it can be demonstrated by assuming a particular case of it, because this would be to assume more than what is neceffary.

Euclid never fuppofed any thing to be possible which he has not before shown to be possible ; but this was not merely to avoid impoffibilities, as fome alledge, but to secure evidence, and to make his reader as certain of his conclufions as he him

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felf was.

PROPOSITIONS.

In the 7th, the literal translation of the enunciation is very
obfcure; there is therefore a little alteration of the expression
introduced, but so as to preserve the same meaning.

Proclus objects to the 12th, that there may be more than one
perpendicular from C to AB, because the circle may cat it in
more than two points; the 17th limits the perpendicular to one,
and therefore the proper place of this iżth is after the 17th.
And it is not used before the 12th of the second book.
To the agth there is added another corollary, which is as fre-

quently

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Book I. quently used as the 14th, and is necessary in the construction of

the 14th and 15th of the tixth book.

in the construction of the 22d, it is now thewn that the circles cut one another, as Dr Simfon has done in his notes.

In the figure to the 24th, it may be shewn, as Dr Simson has done in his notes, that the point F falls below the line EG; or thus: Let DF meet EG in H: and because DHG.

E is the exterior angle of the triangle DEH, 216.1. it is greater a than the angle DEH: but the angle DEH is not b 5. or less bothan DGE, because DG is not less than DE; therefore

the angle DAG is greater than DGH; and DG, that is, DF is € 19. 1. therefore greater cthan DH; wherefore F is below EG.

The 27th may be demonfirated without the help of

E the 16th, thus : Let EF fall on AB and CD, and make the alternate angles AEF, EFD equal; AB is parallel to CD. If not, let them, if posible, meet towards B, D, in G; and make EA equal to

FG, and join AF: and because AE, EF are equal to GF, FE, a 4. 1. and the angle AEF equal to GFE, the base AF is equal a to EG,

and the angle AFE to the angle GEF; and the angle AEF is

equal to EFG; therefore the two angles AFE, EFG are equal b 13. 1. to AEF, FEG, that is, to two right angles b; wherefore AF is

in the same straight line with FG; and the two straight lines 010.AX.1. AFG, AEG Ineet in more than one point, which is impossible d:

Therefore AB and CD do not meet, though produced; where €35. Del 1. fore they are parallel c.

The 29th depends on the 12th axiom, which is not self-evi. dent; and has given much to do, both to ancient and modern geometers. Some have attempted to substitute another axiom in its place; or to give another definition of parallel straight lines; but their attempts have not been successful. Others have attempted to demonstrate the 12th axiom; but when they have done this, by assuming a new axiom, the evidence of their axiom has been objected to; and when they have attempted it without a new axiom, they have hitherto failed. It was therefore thought necessary to demonstrate it without a new axiom ; and since this is the first time that it has been accomplished, it is loped the attempt will be acceptable to accurate geometers, who know the importance of having every thing in geometry eitablished on a solid bases.

PROP

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