APPENDIX (A).

DEFINITIONS AND AXIOMS.

1. THE whole fabric of Geometry rests on a few definitions and axioms, and it is impossible to view, without admiration, the height and solidity of the edifice reared on so narrow a foundation. The nature of the subject-matter of Geometry, viz. Magnitude in all its dimensions, is certainly favourable in the highest degree to the formation of those clear and invariable elementary ideas which are fitted to be the groundwork of science. But the success with which mathematicians are enabled to deduce long trains of consecutive truths, is chiefly due to the fixedness of the ideas which they have to deal with, or, which comes to the same thing, to the definite meaning of the terms employed by them.

2. The abstract ideas with which Geometry is concerned, though easily apprehended, are yet, owing to their great simplicity, in some cases, not susceptible of strict definition; for a definition must be composed of parts, and be resolvable, to use the language of logicians, into a genus and an essential difference. Thus, when we define man to be a rational animal, we refer him to the genus animal, with the essential difference of rationality. In order to define a circle, in like manner, we join to the genus, viz. plane figure, the essential difference, that it is bounded by one line to which all straight lines drawn from a point within the figure are equal. But it is obvious, that when we have to do with simple ideas which cannot be resolved into genus and specific difference, or, in other words, which cannot be referred to, in order to be again distinguished from a class of ideas, definition, properly so called, cannot be employed. We must then have recourse to description, which does not, like definition, determine our

ideas by equivalent expressions, but merely by affixing such marks to them as seem most likely to prevent confusion. The simple ideas, however, which do not admit of being defined, are, in their nature, invariable and distinct, and the least capable of leading to those errors in reasoning, from which the careful avoidance of undefined ideas is in general the best preservative.

3. When we consider a solid body, we perceive at once that it has three dimensions, viz. length, breadth, and thickness: we can also perceive, that it is bounded by a surface or surfaces, in which, abstracting or throwing aside the idea of thickness, we consider only two dimensions, viz. length and breadth. Again, the extremities of surface, or any distances conceived to be measured on it, give us the idea of a line, in which length alone is taken into consideration; ́and furthermore, by continuing our abstraction, and fixing our attention on the extremities of lines, their intersections or particular positions anywise determined on them, we arrive at the idea of a point, which is therefore a mere unit of place, without dimension of any kind, and, consequently, without magnitude.

4. This process of abstraction, however curious it may appear in the descriptions of metaphysicians, is so familiar to the mind, that it generally escapes notice; and it is, at the same time, so easy, that by means of it we are enabled to attain, without effort or meditation, ideas which cannot be conveyed in the complex language of definitions. It is often said, that there exists no such thing in reality as a mathematical point, or as a line, and that these are only the creations of imagination. It might as well be argued, that there exists in the world no such thing as a man abstractedly, or as an animal, or as a living creature. Experience makes us acquainted only with individual existences; but when, by the process of abstraction, we get rid, in each instance, of the individual attributes, and rise to such general conceptions as man, animal, and living creature, who ever supposes that in so doing we go beyond the sphere of reality, or indulge in visions of the imagination? These observations are intended to warn the student not to give way to the belief, that the elementary ideas which form the groundwork of Geometry are peculiarly subtle or unsubstantial, or that the impossibility of strictly

defining them in some cases implies a difficulty of arriving at a just conception of them.

5. It is not the business of the geometrician, however, to explain the origin of those ideas, but merely to fix the meaning of the terms employed to express them; either by definitions properly so called, or, when they are not available, by such descriptions as seem best calculated in each case to secure the learner from error and misconception. Neither is the analytical method by which we form our abstract ideas, proceeding from what is more complex to what is more simple, that best adapted for teaching a science which naturally ascends from what is simple and elementary to that which is complex and abstruse. EUCLID has accordingly followed the synthetic method, commencing with the definition of a point, and proceeding to those of a line and surface: the varieties of bounded surface or figures he goes through in the same order, commencing with the circle which is bounded by only one line, and ascending to those figures which have three, four, or more sides. This method has not allowed him to dissemble the impossibility of defining simple abstractions, and, in consequence, many have felt dissatisfied with his definitions of a point, straight line, &c.; but the importance attached to the defects of those definitions has been hardly less exaggerated than the success with which they are supposed to have been amended.

6. Euclid defines, or rather describes, a point to be “that which has no parts," thereby cautioning his reader, whom he supposes to know what is commonly meant by a point, not to consider it to be divisible. This caution was the more necessary, since the Greek word onμečov, used to express a point, properly signified a mark or dot. The imperfection of this definition, as such, is, however, manifest, and therefore PLAYFAIR has preferred defining a point to be "that which has position without magnitude”—a more specious definition certainly than that of Euclid, but perhaps not more solid, nor better calculated to explain the nature of a mathematical point to those who have not already acquired that idea. In fact, abstract position is an extremely vague idea, nor can it be more appropriately affirmed of a point than of a line. Both must have position-that is to say, some particular position; but the particular position of a point or of a line is

an accidental circumstance, not entitled to enter into the definitions of them; and position in the abstract, applied to the simple ideas of magnitude, means no more than existence; so that to define a point to be "that which has position without magnitude," is equivalent to saying, "that it is (somewhere), without parts ;" that is to say, it expresses no more than what has been already expressed by Euclid, not in so specious a form indeed, but with greater clearness and simplicity.

7. The idea of straightness, the essential character of a straight or right line, is another of those abstractions which, though understood without an effort, and familiar to every mind, is yet not capable of being defined, or of being adequately and directly represented by any combination of words.

A straight line, according to Euclid, is "a line which lies evenly or equally ( ioov) between its extremities." The expression "evenly or equally" stands, it is evident, as much in need of elucidation in this case, as that which it was intended to explain. Any one so ignorant as not to know what is meant by a straight line, could never acquire the idea from the preceding definition. But all who know what is meant by a straight line, can perceive at once that it may be justly said to lie equally between its extremities-that is to say, that it divides the interjacent space equally, leaning neither to the one side nor the other. It was the desire to develope the idea of the evenness or equality peculiar to a straight line, which led to the adoption of the definition given in this volume, which makes the distinction of the straight line to be, that its successive points lie in the same direction. this language it may of course be objected, that by the expression same direction," is only meant "the same right line," and that consequently the terms of the definition need themselves to be defined. But it may be satisfactorily replied, that we are fortunate if, in attempting to define simple ideas, we hit upon terms synonymous with those defined, or equivalent to them, so far as they are applied, and which enable us, in such cases, to attain the ends of definition, much better than can be done by any circumlocution.

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There are many, however, who are not so willing to give over the attempt to define a straight line without the employment of synonymous terms; and among those deemed most

successful is Professor Playfair, whose definition of a straight line runs thus:-"If two lines are such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line."

Now, this is evidently not so much a definition as a criterion presented to the learner from which he is to deduce the essential character of a straight line, and which would lead him to frame the following definition: "a straight line is a line capable of coinciding with a straight line.” Playfair obviously does not overcome, but only eludes, the difficulty of defining a straight line. Nothing can be less consistent with the severe simplicity of the synthetic method, than definitions of the elements, not limited to the exposition of the thing defined considered in itself, but dealing in comparisons, and even anticipating those methods of discovering the relations of magnitudes which properly belong to demonstration. Similar and equal curves are likewise capable of being so applied to each other, that if they meet in any two points, they shall coincide altogether; and though there is certainly a broad distinction between the lines which may, and those which, under the same circumstances, must coincide, yet it ought not to be taken for granted that it is known to those for whom the definition of a straight line is intended.

But Playfair's definition is objectionable, not merely on account of its want of directness, but also of its inaccurate phraseology. Straight lines, he says, are those which, if they coincide (he means, if they meet) in any two points, must coincide altogether. Now, according to the Eighth Axiom, magnitudes which coincide are equal; therefore, it would follow, that all straight lines are equal-an absurd consequence, depending on the ambiguity of the word coincide. Nor is this objection a mere cavilling at terms; for, in divesting Playfair's definition of its equivocal language, we discover, that the considerations on which it rests are more complex than they at first sight appeared to be; and that when reduced to its most correct form, it is nothing more than the converse of Euclid's Tenth Axiom (the corollary of our third definition), viz. two straight lines cannot inclose a space; which is too plainly a deduction from the nature of a straight line, and also too complex (since it involves the consideration of two lines), to be capable of representing that nature or elemental property itself.