between its extreme points, fo as to be related exactly alike to the space on the one fide of it, and to the space on the other, we have a definition that is perhaps a little too metaphyfical, but which certainly contains in it the effential character of a straight line. That Euclid took the definition in this fenfe, 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 employing some reasonings of a more metaphyfical kind than he has any where admitted into his Elements. To fupply the defects of his definition, he has therefore introduced the Axiom, that two ftraight lines cannot inclofe 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 fubftituted another definition of a ftraight line in the room of Euclid's. This definition of a ftraight line was fuggefted by a remark of Bofcovich, who, in his Notes on the Philofophical Poem of Profeffor Stay, fays, Rectam lineam recta congruere totam toti in infini'tum productam fi bina puncta unius binis alterius congruant, patet ex ipfa admodum clara rectitudinis idea quam habemus.' (Supplementum in lib. 3. § 550.) Now, that which Mr Bofcovich would confider as an inference from our idea of ftraightness, feems itself to be the effence 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 expreffing the definition feems 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 straight line, but of ftraight lines. This objection has not much weight in it, yet it cannot be doubted that the definition is more fimple in this latter form.

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

Book I.

Book I. part must not coincide, and therefore by the definition they are not ftraight lines. It follows in the fame way, that two ftraight lines cannot have a common fegment, 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 perufal of a MS. containing many excellent obfervations on the firft Book of Euclid, fuch as might be expected from a philo. fopher diftinguished for the accuracy as well as the extent of his knowledge. He there defined a ftraight line nearly as has been done here, viz. "A ftraight line is that which cannot "meet another ftraight line in more points than one, other. "wife they perfectly coincide, and are one and the fame." Dr Reid alfo contends, that this must have been Euclid's own de finition; because in the first propofition of the eleventh Book, that author argues, "that two ftraight lines cannot have a "common fegment, for this reason, that a straight line does "not meet a straight 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 writings of that geometer ought long fince to have fuggefted 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 fource.

་

There is another characteristic, and obvious property of ftraight lines, by which I have often thought that they might be very conveniently defined, viz. that the pofition of the whole of a straight line is determined by the position of two of its points, in fo much that, when two points of a straight line continue fixed, the line itself cannot change its pofition. It might therefore be faid, that a fraight line is one in which, if the pofition of two points be determined, the pofition of the whole line is determined. But this definition, though it a mounts in fact juft to the fame thing with the former, is ra ther more abstract than it, and is not fo eafily made the foundation of reafoning; fo that, after endeavouring as much as poffible to accommodate it to the understanding of beginners, I have found it beft to lay it afide, and to adopt the definition given in the text.

V.

The definition of a plane is given from Dr Simfon, Euclid's being liable to the fame objections with his definition of a ftraight line, for he says, that" a plane fuperficies is one which "lies evenly between its extreme lines." The defects of this definition are completely removed in that which Dr Simfon has given. Another definition different from both might have been adopted, viz. That thofe fuperficies 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 muft 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 fome advantage; but as the purposes of demonstration cannot be better answered than by that in the text, it has been thought beft to make no farther alteration.

VI.

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 than straight lines. A plane angle is faid to be " the inclina"tion of two lines to one another which meet together, but are "not in the fame direction." This definition is omitted here, because that the angles formed by the meeting of curve lines, though they may become the fubject of geometrical inveftigation, certainly do not belong to the Elements; for the angles. that must first be confidered are those made by the interfection of ftraight lines with one another. . The angles formed by the contact or interfection of a straight line and a circle, or of two circles, or two curves of any kind with one another, could produce nothing but perplexity to beginners, and cannot poffibly be understood till the properties of rectilineal angles have been fully explained. On this ground, without contefting the arguments which Proclus ufes in defence of this definition, I have omitted it. Whatever is not useful, fhould, in explaining the elements of a fcience, be kept out of fight altogether; for, if it does not affift the progrefs of the understanding, it will certainly retard it.

Cc 3

AXIOMS.

Book I.

Book I.

AXIOM S.

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

The 12th Axiom of Euclid is, that If a ftraight line ' meets two straight lines, fo as to make the two interior angles on the fame fide of it taken together lefs than two right angles, thefe ftraight lines being continually produced, shall at length meet upon that fide on which are the angles which are. lefs than two right angles.' Inftead of this propofition, which, though true, is by no means felf evident; another that appeared more obvious, and better entitled to be accounted an Axiom, has been introduced, viz. that two ftraight lines, which interfect one another, cannot be both parallel to the fame ftraight line' On this fubject, however, a fuller explana. tion is neceffary, for which fee 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 refpect to the comparison

of

of triangles. They are demonftrated by what is called the me-
thod of fuprapofition, that is, by laying the one triangle upon
the other, and proving that they muft coincide. To this fome
objections have been made, as if it were ungeometrical to fup-
pofe one figure to be removed from its place and applied to
another figure. The laying,' fays Mr Thomas Simfon in
his Elements, of one figure upon another, whatever evi-
⚫dence it may afford, is a mechanical confideration, and de-
'pends on no poftulate.' It is not clear what Mr Simfon
meant here by the word mechanical; but he probably intended
only to fay, that the method of fuprapofition involves the idea
of motion, which belongs rather to mechanics than geome-
try; for, I think it is impoffible that fuch a Geometer as he
was could mean to affert, that the evidence derived from this
method is like that which arifes from the ufe of inftruments,
and of the fame kind with what is furnished by experience
and obfervation. The demonstrations of the fourth and eighth,
as they are given by Euclid, are as certainly a process of pure
reafoning, depending folely on the idea of equality, as esta-
blished in the 8th Axiom, as any thing in geometry. But,
if still the removal of the triangle from its place be confider-
ed as creating a difficulty, and as inelegant, because it involves
an idea, that of motion, not effential 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 ftraight lines, and if any figure what foever
be conftituted on the one, a figure every way equal to it may
be conftituted on the other. Thus, if AB and DE be two
equal straight lines, and ABC a triangle on the base AB, a
triangle DEF every way equal to ABC may be fuppofed to
be constituted on DE as a base. By this it is not meant to
affert that the method of defcribing the triangle DEF is actual-
ly known, but merely that the triangle DEF may be conceived
to exift in all refpects equal to the triangle ABC. Now, there
is no truth whatsoever that is better entitled than this to be
ranked among the Poftulates or Axioms of geometry; for the
ftraight lines AB and DE being every way equal, there can
be nothing belonging to the one that may not also belong to

the other.

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

Book I.