NOTES. BOOK I. On the Definitions. VARIOUS definitions have been proposed by different writers, to distinguish the straight line, but they may all be shown to be liable to some objection; a circumstance not in the least remarkable, for what is meant by a straight line is so generally understood, that it does not seem possible to convey, by any definition, a better notion of it than the mere mention of its name suggests. Euclid says, "A straight line is that which lies evenly between its extreme points*;" a definition which is both unsatisfactory and useless. Others, following Archimedes, define it as "the shortest distance from one point to another;" but this appears to be assuming too much in a definition, as it immediately leads to the inference, that any two sides of a triangle are together longer than the third side; a proposition which doubtless requires demonstration. The definition which I have given of straight lines, is, I think, as little liable to objection as can be expected. It has, at least, one advantage: it dispenses with Euclid's tenth axiom, viz. "Two straight lines cannot enclose a space;" a property essential to the demonstration of proposition V. of this book. Professor Playfair has defined a straight line as follows: "If two lines are such that they This definition, however, as translated by Mr. Playfair, appears less faulty, viz. "A straight line is that which lies equally between its extreme points,” and in this manner the translation is rendered in the French edition of M. Peyrard. cannot coincide in any two points without coinciding altogether, each of them is called a straight line." This definition is not the best that can be given, for it contains more than is requisite. A definition which involves conditions not absolutely necessary is faulty, as these superfluous conditions may be dispensed with, without leaving the thing defined less distinctly characterized. On this account, Euclid's definition of a square, as having "all its sides equal, and all its angles right angles," has been very properly objected to, as containing superfluous conditions: his definition of an isosceles triangle has, on the other hand, been with equal propriety objected to, as being too restricted; since by defining it as "that which has only two sides equal," the equilateral triangle is excluded. The meaning of Mr. Playfair's definition is, that if two lines which coincide in any two points are always found to coincide throughout their whole extent, each is a straight line. Now it is only necessary that the lines coincide between their coinciding points; for that they will then coincide in every other part may be rigorously demonstrated, as in Prop. V. of these elements. This definition, therefore, is susceptible of restriction. The definition which is given in the text is not liable to this objection; it sufficiently characterizes a straight line, and involves nothing but what must otherwise be assumed as an axiom, viz., that two straight lines cannot include space. Mr. Playfair's definition involves in it, in addition to this, the theorem that "two straight lines cannot have a common segment;" and it is remarkable that that acute geometer should not have perceived that this very circumstance, which in his notes he seems to attribute to the merits of his definition, was in reality a consequence of its defect. I have been thus particular in examining Professor Playfair's definition, because I apprehend that it has hitherto been considered as perfectly unobjectionable, and as possessing the same degree of merit that usually attached to the productions of that distinguished individual. The definition which Euclid has given of an angle is very vague, and can convey but an indistinct notion of angular magnitude; he calls it "the inclination of two straight lines to one another, which meet together, but are not in the same straight line." To understand this definition, it is necessary previously to know what is meant by "the inclination of two straight lines ;" an expression which has not, however, been defined. A modern author of celebrity has endeavoured to give an idea of an angle, by referring to the revolution of a straight line; "A right angle is the fourth part of an entire circuit or revolution of a straight line :" but what an angle has to do with the revolution of a straight line is not easy to conceive; it is certainly not in the smallest degree essential to its existence, for if there were no such thing as a circle, we could quite as readily admit the existence of an angle. The reference of angles to the arcs of a circle is merely an artificial contrivance, adopted for the more convenient measurement and comparison of this class of magnitudes, solely with a view to practical facility; but is in no way connected with the nature of an angle, and is therefore improperly brought forward in its definition. A more usual definition is, "an angle is the opening of two straight lines which meet in a point." By substituting the word between for of, I think this definition becomes more explicit. A perpendicular is generally defined as making equal adjacent angles with the line on which it falls; a definition which appears to require amendment, as it excludes the perpendicular at the extremity of a line. It has therefore been thought proper to make the necessary addition. The definitions of a rhombus, a rectangle, and a square, appear to be rather simpler than those usually given; they involve no more conditions than are absolutely necessary, and those conditions are such as may plainly subsist in the same figure; it being only requisite to admit that one line may be parallel to another, a fact fully established in proposition XII., before either of these definitions are referred to. It may be proper here to remark, that, in the application of terms, I have, in some instances, ventured to depart from ordinary usage. Thus in the comparison of lines, instead of adopting the customary distinction of greater and less, I have preferred the designation of longer and shorter. As a line is understood to be merely length, the terms greater and less appear to be more comprehensive than necessary, and seem to imply other dimensions. For this reason, therefore, they have been changed for terms of more restricted import. I have also confined the term segment to the portion of a circle cut off by its chord, although it has been hitherto applied equally to a portion of a straight line. But this extension of the term appears to be quite unnecessary; for as a line has but one dimension, the expressions part of a line, or portion of a line can convey no ambiguity, and, therefore, on the ground of simplicity, appear to be preferable |