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PROP. II.-INTERC. BOOK.
frivolous and vexatious; though it could not well be omitted if the Second was to be demonstrated.
But in the Second Case, the constant equality of distance between the two points, which is the basis of the possibility of their always occupying the same places during the turning, is a fair specimen of a geometrical reason, and one that might easily go unobserved long after an experimental acquaintance with the truth of the proposition enounced.
PROP. V.-INTERC. BOOK. FIRST CASE. If the surface in which the spheres are assumed to coincide, was an annular one or had a void in the middle, the proof would continue to be applicable; and would be so as long as there was any breadth at all between the inner and outer boundaries of the surface. If there is no breadth, the case merges into that of a self-rejoining line which next follows.
PROP. IX.-INTERC. BOOK. NOMENCLATURE. What is given as the description of a straight line in the translations of Euclid, is nothing but an identical proposition. For to say that a straight line lies evenly' between its extreme points, is only to say that it lies straightly,' or in other words a straight line is a straight line. At the same time this is not exactly what Euclid has said. For the phrase translated evenly,' is in the original et loov, under circumstances of equality.' By which may be understood to be meant, that the line is similarly situated with respect to what may be on each of the two sides of it ; a very darkling description, but not amounting to an identical proposition.
It is at the same time remarkable, that Euclid never afterwards makes
any reference to this ; but when he has occasion for a distinguishing property of straight lines, always has recourse to the property conveyed in what in the translations is called the 10th Axiom, viz. that two straight lines cannot inclose a space. Euclid's description of straight lines therefore practically is, that they are those of which two cannot inclose a space.
Archimedes took as an Axiom, that a straight line is the shortest
between two points* ;' and thence derived the conclusion, that there can be only one straight line between them. This is virtually only a begging of the question, that two sides of a triangle are greater than the third. A parallel case would have been, if a circle had been stated to be the plane figure which incloses a given area under the shortest boundary line.' It might be ultimately true ; but it would be an odd place to start from, for ascertaining the properties of the circle.
Plato declared a straight line to be one in which what is between, is in front of both the endst;' meaning, apparently, that when the two ends are brought into one by the eye, all the intermediate parts are at the same time brought into one with the two ends and with one another. This
appears to be a notification of the fact that rays of light move in straight lines; or at most, a declaration that straight lines are such as examples may be seen of in rays of light. And in this view it seems to have been taken by Proclus, who in quoting it, illustrates it by the occurrence of an eclipse when the moon is in a straight line between the sun and the eyes. A parallel case would have been, if a circle had been stated to be the figure of which examples may be seen in the disks of the sun and full moon.'
Among the new descriptions given by the moderns, that of Bonnycastle is, that a straight line is that which has all its parts lying in the same directions. This is little more than the identical proposition formerly pointed out; for what is a line whose parts do not lie all in the same direction, but a line of which the parts do not all lie in the straight line leading to a particular point or object ? Professor Leslie in his “Rudiments of Plane Geometry ' has given nearly the same description of a straight line; and has built on it as consequences, that no more than one straight line can join two points, and that if a straight line be conceived to turn like an axis about both extremities, its intermediate points
* λαμβάνω δε ταύτα, των τα αυτα πέρατα εχουσών γραμμών ελαχίσης είναι την sidetayo- Archim. De Sphær, et Cyl.
+ και μην ευθύ γε, ου αν το μέσον αμφούν τον εσχάτουν επίπροσθεν η.-Platonis Parmenid.
+ όθεν δή και οι άςρολογικοί φασίν τότε του ήλιου εκλείπειν, όταν επί μιάς ευθείας γένηται αυτός τε και η σελήνη, και το όμμα το ημίσερον. τότε γαρ υπό της σελήνης έπιπροσθείσθαι, μέσης αυτού τε και ημών γενομένης.-Comment. Procli in Ρrimum Euclidis Librum. Lib.2.
§ Bonnycastle's Elements of Geometry. p. 2.
will not change their position*. Professor Playfair's description is, that if two lines are such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line ;' after which he inserts under the title of a Corollary, that 'hence two straight lines cannot inclosé a space. Neither can two straight lines have a common segment; that is, they cannot coincide in part, without coinciding altogethert. The Corollary is a very reasonable inference from the premises ; but the fault is in the premises, which present the fallacy of attempting to wrap up the debateable matter under a name. Coinciding altogether if in any two points,' can no more be called being a straight line, than having the square on the opposite side equal to the sum of the squares of the containing sides can be called being a right angle. No reason is ever given, why the naked fact should exist at all; the inquirer has a right to demand how anybody knows that a line endowed all over with such properties is within the range of possibility,-how it is to be constructed, where it is to be seen,-what evidence there is that the thing pointed out for it, has the properties asserted. If experience (which is experiment) be referred to, experience will equally prove the ratio of the sphere to the circumscribing cylinder ; for whatever is ultimately true, cannot fail to be accordant with experience if tried. But the object of the geometer is to discern reasons.
In the subsequent nomenclature, prolonged appears to be a better term than produced ; if it was only for the advantage of employing the term prolongation, which is so often of use in the practical applications of science. The same terms are afterwards applied to planes ; to which it would be possible to object, plane is not extended in length only, but in length and breadth. The French geometricians, however, have not thought the objection of weight; and it is apprehended that the prolongation of a plane' is a phrase in general use in English.
NOTE TO THE SCHOLIUM.- For an analogy in the case of angles, see Note to Nom. 36, Book I.
• The uniform tracing of a line, which is stretched through its whole extent in the same direction, gives the idea of a straight line. No more than one straight line can hence join two points; and if a straight line be conceived to turn like an axis about both extremities, its intermediate points will not change their position.'— Rudiments of Plane Geometry. p. 19.
+ Playfair's Elements of Geometry. p. 1.
PROP. XII.--INTERC. BOOK. What is given as the demonstration of this Theorem in the Corollary to the 11th Proposition of Simson's Euclid, is null. For if the operation of drawing a perpendicular to the second straight line or to what is assumed to be such, is actually performed according to the directions laid down for it, the result is a second perpendicular distinct from the one previously drawn, and the intended proof falls to the ground.
PROP. XIII.-INTERC. BOOK. NOTE 1. It may occur as an objection here, that it has not
been proved that CL may not be met by the surface of the sphere whose radius is AR, in more points than S. But even if this was the case, it would only follow that every such other point as well as S, must be in the self-rejoining line which is the intersection of the spheres described with the radii AR and BR, and must by the revolution of CL be made to pass through R. By all which, no prejudice would be done to the fact that R is in the surface described
by the revolution of CL. NOTE 2. It may appear as if the demonstration might be
shortened in this place, by making MT turn about M till MT or its prolongation passes through any second point that shall have been assigned, whether in MQOP, or within it, or without. But the proof would fail, if the second point should happen to lie in a straight line which met MQOP in M but not in any other point. Which difficulty, though in a case of necessity it might not be held fatal by a mathematician familiar with the treatment of ultimate values, would be embarrassing to a beginner, and is therefore best obviated by some such process as is given.
NOM. XXXVI.-BOOK I. The thing treated of by geometers under the name of an angle, appears to bear to 'divergence' precisely the same relation that a straight line bears to distance.' It is easy, previously to all mention of straight lines, to assign the meaning of equal distances, by appealing to the capability of coincidence as is done in Nom. XI; but it would not be so easy, clearly to lay down a description of greater or less distances. The nearest perhaps that could
be come to it, would be by propounding some such criterion, as that a greater distance was that where the sphere described with it about a common centre, was exterior to the sphere described with the other; and still there would remain the question how distances, without possessing the aid of anything on which they are to be represented or marked off, are to be added together or subtracted. There wants a substratum of some kind (as the logicians would possibly have termed it), in which the quality of distance shall be embodied, so as to furnish a ready method of comparison by means of addition and subtraction, which are the tests of being what is meant by greater or less; and this want is supplied by the straight line. From the moment therefore that the straight line can be brought upon the scene, all mention of distances should be abolished as belonging to an earlier and inferior state of knowledge; or at all events confined to cases where it is distinctly specified what straight line is intended by the term.
The same analogy may be traced with precision, in the case of divergence
which may be termed 'rotatory distance. It is easy to assign the meaning of equal divergences, by appealing to the fact that the two pairs of straight lines in question can be made to coincide. But for laying down a description of greater or less divergences, there would appear to be no resource unless it were the very clumsy one of saying a greater divergence was that where the cone described by one of the pairs of straight lines turning round one of its legs as a common axis with a common vertex, was exterior to the cone described by the other; and still there would remain the question how divergences, without possessing the aid of anything on which they are to be represented or marked off, are to be added together or subtracted. What is wanted therefore, is the introduction of the plane, to assist, as the straight line did before, by its capability of addition and subtraction. Allow that the thing spoken of shall be a plane surface and not an abstract divergence ;-as it is admitted that the thing spoken of in the other case is a straight line and not an abstract distance ;-and all difficulty vanishes in one case as in the other. Refuse it, and there seems to be induced the same state of obscurity and perplexity, that would be the result of insisting on rejecting the introduction of the straight line into the comparison of distances.
Though the term 'angle’ originally meant a corner, the geometrical idea of an angle equally includes cases where the