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the things at all, and that the result will just as much take place where the exterior angles are avowedly not equal to four right angles. Take, for example, the plane triangle formed by three small arcs of the same or equal circles, as in the margin; and it is manifest that an arc of this circle may be carried round precisely in the way described and return to its old situation, and yet there be no pretence for inferring that the exterior angles were equal to four right angles. And if it is urged that these are curved lines and the statement made was of straight ; then the answer
E is by demanding to know, what property of straight lines has been laid down or established, which determines that what is not true in the case of other lines is true in theirs. It has been shown that, as a general proposition, the connexion between a line returning to its place and the exterior angles having been equal to four right angles, is a non sequitur ; that it is a thing that may be or may not be; that the notion that it returns to its place because the exterior angles have been equal to four right angles, is a mistake. From which it is a légitimate conclusion, that if it had pleased nature to make the exterior angles of a triangle greater or less than four right angles, this would not have created the smallest impediment to the line's returning to its old situation after being carried round the sides ; and consequently the line's returning is no evidence of the angles not being greater or less than four right angles.
14. Franceschini, Professor of Mathematics in the University of Bologna, in an Essay entitled La Teoria delle parallele rigorosamente dimostrata, printed in his Opuscoli Matematichi at Bassano in 1787, offers * a proof which may be reduced to the statement, that if two straight lines make with a third the interior angles on the same side one a right angle and the other an acute, perpendiculars drawn to the third line from points in the line which makes the acute angle, will cut off successively greater and greater portions of the line they fall on. From which it is inferred, that because the portions so cut off go on increasing, they must increase till they reach the other of the two first straight lines, which implies that these two straight lines will meet. Being a conclusion founded on neglect of the very early mathematical truth, that continually increasing is no evidence of ever arriving at a magnitude assigned.
15. A fallacy somewhat more subtle than Franceschini's, though akin to it, may be framed on the consideration of the angle of inter
* See the Notes to Playfair's Elements of Geometry, p. 406; where there is a figure.
section. Let AB, CD be two straight lines in the same plane, making with a third straight line AC the angles CAB, ACD, of which ACD is a right angle and CAB less than a right angle. And to improve the appearance (though this is not indispensable) draw a straight line AE on the other side of AC and in the same plane, making an angle CAE equal to CAB. And from A let a straight line of unlimited length as WX travel along the straight lines AB and AE, cutting AC always at right angles in some point G between A and C. This line will represent Franceschini's succession of perpendiculars. But instead of arguing from its continually cutting off greater and greater portions AG, let it be argued that because it at any tiine makes with AB an angle AHG or BĦX, it may always be removed to a position farther from A without ceasing to cut AB and AE. From which it at first sight might appear to be a reasonable conclusion, that the straight line WX may be carried forward without the possibility of failing to cut AB and AE, till it arrives at C. And the fallacy will perhaps be still more taking, if AB and AE are made to begin by being placed at C, and so are moved from C towards A, as represented by ab and ae ; under which circumstances the allegation that there must always be an angle of some kind at h, has a very inviting appearance as a reason why ab and CD, being continually prolonged, cannot quit one another or fail to meet and make an angle of some magnitude or other, the consequence of which would be that ab and ae might be inoved till a coincides with A and ab with AB, without the possibility of parting company with CD by the way.
The answer to this is by inquiring, whether there are no lines in which there may be the same kind of evidence on the subject of the angle, but where it is certain that a straight line as WX cannot be carried on to an unlimited extent as proposed. And here it is easy to show that there may. Take, for example, any hyperbola, and from the vertex draw a perpendicular to each of the asymptotes ; and let the two halves of the linear hyperbola, together with the perpendiculars and the portions of the asymptotes cut off by them on the side remote from the intersection of the asymptotes, be placed so that the perpendiculars shall coincide and the asymptotes in consequence be in one straight line, as ST in the figure below. Upon which it is clear, that however it may be pleaded that there may always be an angle smaller than AHG or BHX
between WX and AB, WX cannot be carried beyond the line of the asymptotes ST without ceasing to meet AB; and consequently cannot be carried till it meets CD, if CD lies on the other side of ST as represented in the figure.
It follows therefore, that to say there will always be the possibility of a further diminution of the angle, is not enough. It is the sophism of Achilles and the tortoise; which argued that because after running a mile, half a mile, a quarter of a mile, &c. Achilles would always be behind by the last-mentioned fraction of a mile, he would never overtake or pass the tortoise. The solution resolving itself into the fact, that these quantities though infinite in number are finite and surpassable in amount.
To establish the union of the lines to any particular extent that may be desired, it is consequently necessary to prove, not only that the angle at the intersection is capable of diminution, but that the angle each way (that is to say, both the angle AHG and the angle GHB) shall never be reduced to less than some given angle. Which is what is done accordingly, in the Proposition numbered XXVIII D of the present work.
16. In a tract entitled “ The Theory of Parallel Lines perfected; or the Twelfth Axiom of Euclid's Elements demonstrated. By Thomas Exley, A. M.-London. Hatchard. 1818.'—the proof rests on taking for granted (in the Second Proposition) that if four equal straight lines in the same plane, making right angles with one another successively towards the same hand, do not meet and inclose a space, a fifth if prolonged both ways must inevitably accomplish it. A conclusion which may be resolved into taking for granted that the three angles of a rectilinear triangle are greater than a right angle and a half; for if they were equal to this, the angles of an equilateral and equiangular oktagon would be right angles, and the fifth straight line in the series proposed would never meet the first ; still more if they were less. And in the same manner if it was urged that a sixth, seventh, &c. perpendicular must meet the first straight line, it would only resolve itself into a demand for adınitting without proof, that the three angles of a triangle are greater than some other amount capable of being specified. There is no obscurity about the fact that four such straight lines, and still more five, are found on experiment to meet; but the object was to discover why they necessarily meet. And between the observed fact and the explained fact, there is a difference of the same kind as between Kepler's observation of the proportion between the periodic times and distances of the planets, and Newton's explanation of the cause.
17. The demonstration presented by M. Legendre in the earlier editions of his · Eléments de Géométrie,' consisted in first establishing that the three angles of a rectilinear triangle cannot be greater than two right angles (which may be passed over as irrefragable and liable to no remark), and afterwards proceeding to show cause why they should not be less. But the evidence offered on this latter point, depended on taking for granted that two straight lines (DE and BE in fig. 35 a in the Fourth Edition, and probably in the subsequent editions as far as the Eighth inclusive) meet when they make with a third straight line (DB) angles of which one (us EDB) is, or may be made to be, less than a right angle, and the other looks less than a right angle, but without further evidence. In the Seventh
Edition an attempt was made to show that the lines must meet; but the proof advanced involves the same fallacy as that of the Bolognese Professor*.
18. This demonstration was withdrawn in the Ninth Edition, and a new one inserted in the Twelfth. The new one depended upon taking in any triangle an angle that is not lesst than any other in the triangle, and a second that is not greater (See Douzième édition p. 20, and Plate); bisecting the side opposite to the second angle, and drawing a straight line from the angular point to the point of bisection ; cutting off in this straight line and its prolongation a part from the angular point equal to the side opposite to the firstmentioned angle (viz. that angle which is not less than any other in the triangle), and in this side and its prolongation towards the same hand a part equal to double the straight line between the angular point and the point of bisection formerly mentioned, and joining the extremities of the two parts thus cut off. It is not difficult to show, that in the new triangle thus last constructed, the sum of the three angles is the same as in the original triangle ; and moreover that of the angles of these two triangles which are at a common point, that belonging to the new triangle is not greater than balf that of the old, while another of the angles of the new triangle is equal to their difference. And if these operations be applied in like manner to the last constructed triangle, a third triangle will be constructed having the same relations to the second ; and so on. Whence it follows, that the described process may be continued, till two of the angles of the last-resulting triangle are together less than any magnitude that shall have been assigned ; and consequently the third or remaining angle may be made to approach, within any magnitude however small it may be chosen to assign, to the sum of the three angles of the original or any of the intervening triangles.
All this is irrefragable, but not so the proposition next taken for granted, which is that the third angle last-mentioned approaches within any magnitude however small it may be chosen to assign, to the sum of two right angles. That it approaches it (that is that the angle continually grows larger) is certain ; but that it approaches to it within any magnitude however small, is the point which, as in so many parallel instances, is taken for granted without sufficing proof. The weakness in the actual case, is in the fact that the base or side opposite to the continually increasing angle, becomes itself of unlimited length. If the resulting triangles
had been all on the same base, the inference might perhaps have been conceded to be good. But it is precisely because by the extension of the base to an unlimited magnitude the progress of the operation is removed from human eyes, that the force of the inference is diluted and done
Just as fast as the diminution of the two acute angles appears to induce a necessity for the obtuse angle's approximating to the sum of two right angles, does the increase of the length of the sides hold forth an augmented probability that the angle may after all evade increasing by the quantity required to make it attain to two right angles in the
* Elém. de Géom. Par A.M. Legendre, 7ème édit. p. 280. Note II.
+ Not less and not greater are substituted for the greatest and least of the original, from a persuasion that these last are an oversight. The demonstration is plainly intended to be applicable to any triangle ; but the terms used would not apply to an equilateral triangle, or any kind of isoskeles.
end. To argue that when the acute angles are nothing, or the lines coincide, the third angle will make a straight line,-is substituting for what really happens, what by the hypothesis is never to happen. The demonstration is therefore finally of the same strength as Franceschini's and others that have been mentioned. There is evidence of a perpetual approach towards a given magnitude; but there is not evidence of the degree and rapidity of approach which are neces. sary to ensure arriving at it.
19. Another demonstration, or step towards a demonstration, presented by the same author (See Note II. p. 279, 12ème édition), consists in representing, that if any angle less than two right angles is bisected, all perpendiculars to the bisecting straight line must meet the sides, because otherwise there would be a straight line shut up between the lines that make an angle, 'which is repugnant to the nature of the straight line.'. On which it is sufficient to observe, that the existence, cause, and origin of this repugnance, are precisely what it was in question to establish.
20. The next paragraph in the same page is directed to establishing the sort of postulate assumed in the last, viz. that a straight line cannot be shut up within an angle. The argument appears to be, that either of the straight lines which make an angle, being prolonged will divide the infinite plane in which it exists into equal parts, and any other straight line must do the same; but a straight line that should be shut up within the angle, would cut off more on one side and less on the other; therefore a straight line cannot be shut up within an angle. Whoever examines this closely, will see that it would equally prove that two straight lines cannot be parallel to one another; for in that case it might equally be urged, that if the one cuts the plane in halves, the other must cut off more on one side and less on the other. The whole is manifestly a mistake arising from overlooking Plato's observation, that equality of magnitude can only be predicated of things finite.
21. The next in order is the so-called analytical proof, which professes to demonstrate that if two angles in one rectilinear triangle are respectively equal to two in another, the remaining angles are necessarily equal. "If two angles of a triangle and the side between them are given, the rest of the sides and angles of that one triangle are determined ; that is to say, they can severally be only of one certain magnitude and no other. Hence, said the advancers of this demonstration, the angle opposite to the given side is a function of the two angles and the given side ;-their meaning by this term being, that a quantity is a function of other quantities, when on those other quantities being fixed and determined in magnitude, the first quantity is necessarily fixed and determined in magnitude, or is what Euclid in his Book of Data would call given. Let the right angle be equal to unity or 1, and then the angles will all be numbers somewhere between 0 and 2; and since the third angle is a function of the two other angles and the side between them, it will follow that the side cannot enter as an element into the determination of the magnitude of the angle.' And this, they said, because the side is heterogeneous with the other quantities which are numbers, and no equality can be compounded or made to exist between them*.
• 'Il faut donc que l'angle C soit entièrement déterminé, lorsqu'on connait