Book I. and fecond poftulates, for we certainly understand by these poftulates, that only one ftraight 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 pofition 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 faid, and because it is his 26th propofition of the Data, which is the 29th of Simfon's edition. Befides, it gives too remote an intimation of the nature of ftraight lines, and their properties cannot be eafily deduced from it.

The other definition is, that a straight line is that which cannot cut another ftraight line in more points than one. In the Greek, this is given in the firft propofition of the eleventh book, as the reason why two ftraight lines cannot have a common fegment; and this laft is affumed every where in the Elements. Their cutting only in one point feems to be a particular case, or at least a confequence 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, because, when we are not actually applying ftraight lines to one another, it may be found that two points of one of them coincide with two points of another, in which cafe 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 difadvantages, for then the only criterion of the coinciding of ftraight lines, would be their meeting in two points: To fuppofe that they coincide, without firft finding that they meet in two points, would be to affume a property of them, not contained in the definition: We are not therefore warranted to fay, as in the fourth propofition, Let the triangle ABC be applied to DEF, fo that the point A may be on D, and the ftraight line AB upon DE; for all that follows from AB and CD being ftraight lines, according to this definition, is, that they would coincide, if, befides A being on D, fome other point of AB be shown to coincide with fome other point of DE. Farther, the property contained in this definition is not the fame with the principle upon which the mechanical defcription 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 as the inclination of lines: Apollonius called it cuvaywyn, the draw,

ing together of a fuperficies to a point; and others called it the Book I. first distance of the containing lines: And the philofophers difputed whether it was a quantity, or a quality, or a relation, or whether it belonged to one or to feveral of the categories. The moderns have 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 metaphyfical reafoning. Thus we find, that nothing in the Elements depends upon thefe nine definitions; and therefore the purpofe of inftruction would be better obtained by removing them from the definitions, and explaining them in an introduction.

fo

Befides thefe, there are several other definitions which might have been omitted, or altered for the better. Thus, in the definition of an equilateral triangle, there is nothing peculiar to a triangle, for every rectilineal figure is in the Elements faid 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 faid to be neceffary for completing the arrangement of triangles; but even this cannot be pleaded for the definitions of quadrilaterals, for the claffification of them in the Elements is complete, and altogether different from that in the definitions; so that the definitions are not well adapted to the work itself, and might have been altered with advantage.

It fhall only be farther remarked, that the whole defign of definitions is, to give names to things that have certain properties; and that unless the properties afcribed 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 propofitions, fuppofe the existence of the things concerned in them. Accordingly, Euclid either affumes 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.

AXIOM S.

THE only alteration that is made here, is in the roth axiom,

which is now made more extenfive than it was before.

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

the

Book I. the Elements, though it be no where repeated, except in the first propofition of the eleventh book.

For example, it is affumed in the second poftulate; for, as Proclus obferves, that poftulate would be nugatory, if the terminated straight line could be the common fegment of two ftraight lines. In like manner, as Zeno objects, the first propofition is not fufficiently demonftrated, without affuming, that, two flraight lines cannot have a common fegment: For if the ftraight lines AC, CB can have a common fegment, the fides 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; fo that the triangle may not be equilateral. Likewife, without this fuppofition, the fourth would not be fufficiently demonstrated; for when AB is applied to DE, if they could have a common fegment lefs than AB, they would coincide the length of this fegment; but not afterwards; fo that the point B might not coincide with the point E.

The fifth depends on the fame fuppofition; for if either fide of the ifofceles triangle, as AB, be the common fegment of two ftraight lines, thefe lines would make unequal angles with the bafe BC on the other fide of it: But each of them would be equal to the angle BCE, by the fifth prop. In the fame manner, it may be fhown, that the fame thing is fuppofed in many other propofitions; and therefore it, is evident that Euclid confidered it as an axiom.

As to axioms, it is generally allowed, that they ought to be felf-evident, and likewife as few as poffible; for nothing ought to be confidered as felf-evident that is capable of demonstration.

The poftulates ought likewise to be as few as poffible; for, as Sir Ifaac Newton obferves, poftulates are principles which geometry borrows from the arts, and its excellence confists in the. paucity of them. The poftulates of Euclid are all problems derived from the mechanics; but there are things affumed by geometers, which are neither felf-evident nor problems; fuch as that called the 12th axiom: and it is plain, that fuch affumptions ought not to be made, but when the thing affumed is demonftrated elsewhere, or when it cannot poffibly be demonftrated; for all certainty arifes from felf-evidence. Poftulates likewife 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 affumer must define it; otherwife 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 fame opening of the compaffes. And because a straight line is drawn either by ftretching a thread, or by applying another

ftraight

traight 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 ftretched thread, which is its invariably affuming the fame pofition, as often as it is stretched in the fame fituation.

Many of the moderns complain, that Euclid, by taking fo little for granted, has narrowed the foundation of geometry too much. And they maintain, that we are at liberty to affume any thing, the existence of which does not imply a contradiction, or involve an impoffibility. But were this the cafe, we might affume every thing, for we certainly fuppofe nothing impoffible, 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 poffible; but many things may appear evident to a writer, which are not fo to a learner. An author of Elements. ought to confider the ftate of his reader's mind, who, being altogether unacquainted with geometry, can admit nothing to be felf-evident that is peculiar to it. It is neceffity, and not liberty, which is the cause of affumptions; and the writer who affumes what he is not obliged to affume, is not writing for his reader's inftruction, but is endeavouring to beg fame, by humouring his reader's indolence.

It follows from this, that affumptions ought to be as particu lar as poffible; for example, we ought not to affume a general propofition, if it can be demonftrated by affuming a particular cafe of it, because this would be to affume more than what is neceffary.

Euclid never fuppofed any thing to be poffible which he has not before fhown to be poffible; but this was not merely to avoid impoffibilities, as fome alledge, but to fecure evidence, and to make his reader as certain of his conclufions as he himfelf was.

PROPOSITIONS.

IN the 7th, the literal tranflation of the enunciation is very obfcure; there is therefore a little alteration of the expreffion introduced, but fo as to preferve the fame 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 11th is after the 17th. And it is not used before the 12th of the fecond book.

To the 15th there is added another corollary, which is as frequently

Na

Book I. quently ufed as the 14th, and is neceffary in the conftruction of the 14th and 15th of the fixth book.

a 16. 1.

18. I.

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

C

D

E

H

G

F

E B

In the figure to the 24th, it may be fhewn, as Dr Simfon has done in his notes, that the point F falls below the line EG; or thus: Let DF meet EG in H: and becaufe DHG. is the exterior angle of the triangle DEH, it is greater a than the angle DEH: but the angle DEH is not b 5. or less than DGE, becaufe DG is not lefs than DE; therefore the angle DBG is greater than DGH; and DG, that is, DF is € 19. 1. therefore greater than DH; wherefore F is below EG. The 27th may be demonftrated without the help of 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 poffible, meet towards B, D, in G; and make EA equal to FG, and join AF: and because AE, EF are equal to GF, FE, and the angle AEF equal to GFE, the bafe AF is equal 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 ; wherefore AF is in the fame ftraight line with FG; and the two straight lines d10.Ax.1. AFG, AEG meet in more than one point, which is impoffible : Therefore AB and CD do not meet, though produced; where€35.Def.1. fore they are parallel .

a 4. 1.

2

The 29th depends on the 12th axiom, which is not self-evident; and has given much to do, both to ancient and modern geometers. Some have attempted to fubftitute another axiom in its place; or to give another definition of parallel ftraight lines; but their attempts have not been successful. Others have attempted to demonftrate the 12th axiom; but when they have done this, by affuming 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 neceffary to demonftrate it without a new axiom; and fince this is the first time that it has been accomplished, it is hoped the attempt will be acceptable to accurate geometers, who know the importance of having every thing in geometry eftablished on a folid bases.

PROP,