66 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 bis demonstrations are founded. As this manner of preceeding 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 productum si bina puncta unius binis alterius congruant, patet ex ipsa admodum clara rectitudinis idea quam habemus." (Supplementum in lib. 3. $ 550.) Now, that which Mr. Boscovich would consider as an ioference 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. On this principle we have given the definition above, If there be two lines which cannot coincide in two points, without coinciding altogether, each of them is called a straight line. This definition was otherwise expressed in the two former editions : it was said, that lines are straight lines which cannot coincide in part, without coinciding altogether. This was liable to an objection, viz. that it defined straight lines, but not a straight line ; and though this in truth is but a mere cavil, it is better to leave no room for it. The definition in the form now given is also more simple. From the same definition, the proposition which Euclid gives as an Axiom, 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 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 first Edition, 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 philosopher distinguished 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, otherwise they perfectly coincide, and are one and the s6 same.” Dr. Reid also contends, that this must have been Euclid's own definition ; because in the first proposition of the eleventh Book, that author argues, "that two straight lines cannot have a common seg“ment, 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 a that the writings of that geometer ought long since to have suggested this definition to his commentators ; and it reminds me, that I might have learned from these writings what I have acknowledged above to be derived from a remoter source. There is another characteristic, and obvious property of 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 said, 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 amount in fact to the same thing with that already given, is rather more abstract, and not so easily made the foundation of reasoning. I therefore thought it best to lay it aside, and to adopt the definition given in the text. 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. 1 a 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 the other lines than straight hnes. A plane angle is said to be “ the inclination of two lines to one another which meet together, but are not in the same direccó tion.” This definition is omitted here, because that the angles formed by the meeting of curve lines, though they may become the subject of geometrical investigation, 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 nothing but perplexity to beginners, and cannot possibly be understood till the properties of rectilineal angles have been fully explained. On this ground, I am of opinion, that in an elementary treatise, it may fairly be omitted. Whatever is not useful, should, in explaining the elements of a science, be kept out of sight altogether ; for, if it does not assist the progress of the understanding, it will certainly retard it. AXIOMS. 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 10th. 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 angles, 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 proposition, which though true, is by no means selfevident; another that appeared more obvious, and better entitled to be accounted an Axiom, has been introduced, viz. “ that two straight “ lines, which intersect one another, cannot be both parallel to the “ same straight line." On this subject, however, a fuller explanation is necessary, for which see the note on the 29th Prop. PROP. IV. and VIII. B. I. The fourth and eighth propositions of the first book are the foundation of all that follows with respect to the comparison of triangles. They are demonstrated by what is called the method of supraposition, 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 suppose one tigure 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 consideration, and depends 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 supraposition involves the idea of motion, which belongs rather to mechanics than geometry ; for I think it is impossible that guch a Geometer as he was could mean to assert, that the evidence derived from this method is like that which arises from the use of instruments, 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 reaoning, depending solely on the idea of equality, as established in the 3th Axiom, as any thing in geometry. But, if still the removal of the triangle from its place be considered 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 postulates, we be allowed to add the following, viz. That if there be two equal straight lines, and if any figure whatsoever be constituted on the one, a figure every way equal to it may be constituted 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 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 actually 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 can be nothing belonging to the one that may not also belong to the other. On the strength of this postulate the fourth Proposition is thus demonstrated. If ABC, DEF be two triangles, such that the two sides AB and AC of the one are equal to the two ED, DF of the other, and the angle, BAC, contained by the sides AB, AC of the one, equal to the angle EDF, contained by the sides ED, DF of the other; the triangles ABC and EDF are every way equal. А D G B E I On AB let a triangle be constituted every way equal to the triangle DEF; then if this triangle coincide with the triangle ABC, it is evident that the proposition is true, for it is equal to DEF by hypothesis, and to ABC, because it coincides with it ; wherefore ABC, DEF are. equal to one another. But if it does not coincide with ABC, let it have the position ABG; and first suppose G not to fall on AC; then the angle BAG is not equal to the angle BAC. But the angle BAG is equal to the angle EDF, therefore EDF and ABC are not equal, and they are also equal hy hypothesis, which is impossible. Therefore the point G must fall upon AC ; now, if it fall upon AC but not at C, then AG is not equal to AC ; but AG is equal to DF, therefore DF and AC are not equal, and they are also equal by supposition, which is impossible. Therefore G must coincide with C, and the triangle AGB with the triangle ACB. But AGB is every way equal to DEF. therefore ACB and DEF are also every way equal. Q. E. D. By help of the same postulate, the 5th may also be very easily demonstrated. Let ABC be an isosceles triangle, in which AB, AC are the equal sides, the angles ABC, ACB opposite to these sides are also equal. Draw the straight line EF equal to BC, and suppose that on EF the triangle DEF is constituted every way equal to the triangle ABC, that is, having DE equal to AB, DF to AC, the angle EDF to the angle BAC, the angle ACB to the angle DFE, &c. AA B F Then, because DE is equal to AB, and AB is equal to AC, DE is equal to AC ; and for the same reason, DF is equal to AB. And because DF is equal to AB, DE to AC, and the angle FDE to the angle BAC, the angle ABC is equal to the angle DFE, (4. 1.). But the angle ACB is also, by hypothesis, equal to the angle DFE; therefore the angles ABC, ACB are equal to one another. Q. E. D. Thus also, the 8th proposition may be demonstrated independently of the 7th. Let ABC, DEF be two triangles, of which the sides AB, AC are equal to the sides DE, DF each to each, and also the base BC to the the base EF ; the angle BAC is equal to the angle EDF. А. D B C F |