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between its extreme points, so as to be related exactly alike Book I. to the space on the one side of it, and to the space on the other, we have a definition that is perhaps a little too metaphysical, but which certainly contains in it the essential character of a straight line. That Euclid took the definition in this sense, however, is not certain, because he has not attempted to deduce from it any property whatsoever of a straight line, and indeed, it should seem not easy to do fo, without em. ploying some reasonings of a more metaphysical kind than he has any where admitted into his Elements. To supply the defects of his definition, he has therefore introduced the Axiom, that two straight lines cannot inclose a space, on which Axiom it is, and not on his definition of a straight line, that his demonstrations are founded. As this manner of proceeding is certainly not so regular and scientific as that of laying down a definition, from which the properties of the thing defined may be logically deduced, I have substituted another definition of a straight line in the room of Euclid's. This definition of a straight line was suggested by a remark of Boscovich, who, in his Notes on the Philosophical Poem of Professor Stay, says, ' Rectam lineam' rectæ congruere totam toti in infinitum productam fi bina puncta unius binis alterius congruant,

patet ex ipfa admodum clara rectitudinis idea quam habe*mus.' (Supplementum in lib. 3. $ 550.) Now, that which Mr Boscovich would consider as an inference from our idea of straightness, seems itself to be the essence of that idea, and to afford the best criterion for judging whether any given line be straight or not. It may, however, be better to express the definition, a little differently from. Def. III., as given above, thus : If there be two lines which cannot coincide in two points, without coinciding altogether, each of them is called a straight line. This

way of expressing the definition seems preferable to that in the text, as it has been objected, that in the form in which it stands there, it is a definition not of a fraight line, but of straight lines. This' objection has not much weight in it, yet it cannot be doubted that the definition is more fim. ple in this latter form.

From this definition the Axiom above mentioned, viz. That two straight lines cannot inclose a space, follows as a necessary consequence. For, if two lines inclose a space, they must intersect one another in two points, and yet in the intermediate

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Book I. part must not coincide, and therefore by the definition they

are not straight lines. It follows in the same way, that two straight lines cannot have a common segment, or cannot coincide in part, without coinciding altogether.

After laying down the definition of a straight line, as in the text, I was favoured by Dr Reid of Glasgow with the perusal of a MS. containing many excellent observations on the first Book of Euclid, such as might be expected from a philo. sopher diftinguished for the accuracy as well as the extent of his knowledge. He there defined a straight line nearly as has been done here, viz. “ A straight line is that which cannot “ meet another straight line in more points than one, other. “ wise they perfectly coincide, and are one and the same.” Dr · Reid also contends, that this must have been Euclid's own definition ; because in the first propofition of the eleventh Book, that author argues, " that two straight lines cannot have a “ common segment, for this reason, that a straight line does “ not meet a Itraight line in more points than one, otherwise “ they coincide." Whether this amounts to a proof of the definition above having been actually Euclid's, I will not take upon me to decide ; but it is certainly a proof that the wri. tings of that geometer ought long since to have suggested this definition to his commentators; and it reminds me, that I might have learnt from these writings what I have acknowledged above to be derived from a remoter source.

There is another characteristic, and obvious property straight lines, by which I have often thought that they might be very conveniently defined, viz. that the position of the whole of a straight line is determined by the position of two of its points, in so much that, when two points of a straight line continue fixed, the line itself cannot change its position. It might therefore be faid, that a straight line is one in which, if the position of two points be determined, the position of the whole line is determined. But this definition, though it amounts in fact just to the same thing with the former, is ra. ther more abstract than it, and is not so easily made the foundation of reasoning ; so that, after endeavouring as much as possible to accommodate it to the understanding of beginners, I have found it best to lay it aside, and to adopt the definition given in the text.

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Book I.


The definition of a plane is given from Dr Simson, Euclid's being liable to the same objections with his definition of a straight line, for he says, that -- a plane superficies is one which “ lies evenly between its extreme lines.” The defects of this definition are completely removed in that which Dr Simson has given. Another definition different from both might have been adopted, viz. That those superficies are called plane, which are such, that if three points of the one coincide with three points of the other, the whole of the one must coincide with the whole of the other. This definition, as it resembles that of a straight line, already given, might, perhaps, have been introduced with some advantage ; but as the purposes of demonstration cannot be better answered than by that in the text, it has been thought best to make no farther alteration.


In Euclid, the general definition of a plane angle is placed before that of a rectilineal angle, and is meant to comprehend those angles which are formed by the meeting of other lines thán straight lines. A plane angle is said to be " the inclina. “tion of two lines to one another which meet together, but are “not in the same direction.” This definition is omitted here, because that the angles formed by the meeting of curve lines, though they may become the subject of geometrical invefti. gation, certainly do not belong to the Elements; for the angles that must first be considered are those made by the intersection of straight lines with one another. The angles formed by the contact or intersection of a straight line and a circle, or of two circles, or two curves of any kind with one another, could produce pothing but perplexity to beginners, and cannot poffibly be understood till the properties of rectilineal angles have been fully explained. On this ground, without contesting the arguments which Proclus uses in defence of this definition, I have omitted it. Whatever is not useful, should, in explaining the elements of a science, be kept out of fight altogether; for, if it does not aslist the progress of the underlanding, it will certainly retard it.


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Book I.


AMONG the Axioms there have been made only two alterations. The 10th Axiom in Euclid is, that two straight lines • cannot inclose a space ;' which having become a corollary to our definition of a straight line, ceases of course to be ranked with self-evident propositions. It is therefore removed from among the Axioms, and that which was before the 11th is accounted the roth,

The 12th Axiom of Euclid is, that " If a straight line • meets two straight lines, so as to make the two interior angles

on the same side of it taken together less than two right an*gles, these straight lines being continually produced, shall at • length meet upon that side on which are the angles which are • less than two right angles. Instead of this propofition, which, though true, is by no means self evident; another that

appear. ed more obvious, and better entitled to be accounted an Ar. iom, has been introduced, viz. “ that two straight lines, which interfect one another, cannot be both parallel to the fame straight line." On this subject, however, a fuller explana. tion is necessary, for which see the note on the 29th Prop.

PROP. IV. and VIII. B. I.

The fourth and eighth propofitions of the first book are the foundation of all that follows with respect to the comparison


Book I.

of triangles. They are demonstrated by what is called the me-
thod of suprapofition, that is, by laying the one triangle upon
the other, and proving that they must coincide. To this some
objections have been made, as if it were ungeometrical to fup-
pose one figure to be removed from its place and applied to
another figure. . The laying,' says Mr Thomas Simson in
his Elements, of one figure upon another, whatever evi.
. dence it may afford, is a mechanical conlideration, and de-
• pends on no postulate. It is not clear what Mr Simson
meant here by the word mechanical ; but he probably intended
only to say, that the method of fuprapofition involves the idea
of motion, which belongs rather to mechanics than geome-
try; for, I think it is impossible that such a Geometer as he
was could mean to affert, that the evidence derived from this
method is like that which arises from the use of inftruments,
and of the same kind with what is furnished by experience
and observation. The demonstrations of the fourth and eighth,
as they are given by Euclid, are as certainly a process of pure
reasoning, depending solely on the idea of equality, as efta-
blished in the 8th Axiom, as any thing in geometry. But,
if ftill the removal of the triangle from its place be consider-
ed as creating a difficulty, and as inelegant, because it involves
an idea, that of motion, not essential to geometry, this defect
may be entirely remedied, provided that, to Euclid's three
poftulates, we be allowed to add the following, viz. That if
there be two equal straight lines, and if any figure what foever
be constituted on the one, a figure every way equal to it may
be constituted on the other. Thus, if AB and DE be tivo
equal straight lines, and ABC a triangle on the base AB, a
triangle DEF every way equal to ABC may be supposed to
be constituted on DE as a base. By this it is not meant to
assert that the method of describing the triangle DEF is actual-
ly known, but merely that the triangle DEF may be conceived
to exist in all respects equal to the triangle ABC. Now, there
is no truth whatsoever that is better entitled than this to be
ranked among the Postulates or Axioms of geometry; for the
straight lines AB and DE being every way equal, there car
be nothing belonging to the one that may not allo belong to
the other.

On the strength of this postulate the fourth Prop. is thus demonstrated.

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