STRAIGHT, STRAIGHT LINE, PLANE. There is no occasion to define a straight line as matter of information; so that we have here only to consider the definitions which have been given and their relative merits, taking them as attempts to produce a mathematical description of straightness.
There are three attempts at definition of a straight line ; by Plato (or one of his immediate school), by Archimedes (as is said), and by Euclid. The moderns have repeated these various forms, but have not, to our knowledge, ever succeeded in producing a definition entirely new which did not contain the defects of one or other of the three just mentioned.
The Platonic definition, according to Proclus, is as follows :—"A straight line is that of which the middle parts hide (1,rorpoa0(1) the extremities ; " a physical definition, owing its truth to the' circumstance of the rays of light proceeding in straight lines, and involving the notion of straightness as a part of its own explanation. This defini tion has been little if at all used by geometrical writers.
Archimedes defines a straight line as the shortest distance between two points, or at least this definition is often attributed to him, but not correctly. It is one of his postulates iu the book on the Sphere and Cylinder, that of all lines drawn between two points the least is that which is straight : but he is too well judging a geometer to assign such a property as a definition. The Arabs substituted the shortest distance description for the definition in Euclid, and accordingly our earlier editions of Euclid do the same; nor was this flaw removed until 1505, when Zamberti translated Euclid from the Greek. It has often been supposed that this shortest-distauce definition is good as a definition, though not proper for a pupil in geometry, an opinion from which we must dissent : for how is it knows to those who are yet to learn what a straight line is, whether there can be a shortest distance ? That is, how is it known that there are not many distances between two points, on different lines, which are severally shorter than any other distance, and equal to one another ? The answer is, no doubt, that the mind has a perfect conception of the impossibility of such a thing ; and the rejoinder is—yes, because the miud has a perfect con ception of a straight lino : that is to say, the definition is only saved from causing confusion by its own uselessness. Again, the supposition
that measurement of distances on all manner of curves is to be a pre liminary to one of the definitions of a science which treats no curve but the circle, and does not succeed, by reason of certain limitations of process, in measuring distance even on that one, is an incongruity.
Euclid defines a straight line to be that which lies evenly (4 fo-ou Kano) between its extreme points. The words d( tow have been trans lated ex (equo by Barocius, ex aquali by Zamberti, equally by Billingsley (taking some of the oldest translations as specimens). The definition wants precision, but the meaning is obvious. Two points being given, the surrounding space may be viewed in all manner of relations to those two points, as above or below, right or left, &c. The straight line which joins the two points is that which is not more related to one of these notions than to any other; and throughout its whole length takes au even course, without a possibility of being claimed, so to speak, by any one of the surrounding parts of space rather than by any other.
In makinis such a definition Euclid is well aware that he cannot rest any conclusion upon it, and that in the postulate that two straight lines cannot inclose a space lies all his power of producing a theorem. Why then, it may be asked, does ho introduce a definition at all f Why not give the nudes to understand that a straight line is a notion tint vensally understood and incapable of definition in simpler tertna f To these questions the answer may be twofold. In the first place, he is not answerable for the genius of any language but his own, and it is very possible that to a Greek commencing geometry, 41,047a might be a hard word, and 1/ 7Tal a real explanation ; in which case his definition is defensible until it can be shown that ho might have chosen a better one. We are not to judge of the force of the last-quoted words from the ex alio of the middle Latin, or the evenly or equally of the English. Secondly, he is evidently, in sonic of the first defini tions, recalling, and not instilling, notions: ho is proceeding with his reader as by words to which both attach a conception, and he tries these words for use by ascertaining that both parties agree on such circumlocution as can be substituted for them.