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In this the reference is to what the analysts had denominated the principle of homogeneity; a principle in itself irrefragable, but like all others, capable of being ill applied. Wherever quantities are to be equal, they must be homogeneous or of the same kind; for equality is nothing but the capability of coincidence, and things heterogeneous cannot coincide. A mile of length, or two, or three, or four miles, can by no possibility be equal to an hour of time; the assertion would be ipso facto foolish and unmean
four miles ten hours ing. But there is no objection to saying that
two miles five hours: because the first of these expressions means only the number of times that the quantity two miles can be taken in the quantity four miles, which is the number two; and, five hours may be taken the same number of times in ten. And by the same rule, there is no
ten hours objection to saying that four miles = two miles x
five hours this means nothing but that four miles = two iniles x the number which results from seeing how often five hours can be taken in ten. It follows therefore that heterogeneous quantities enter equations by pairs ; or at all events are reducible to pairs by running some two or more of them into one by the operation of addition or subtraction. There cannot be the slightest idea of questioning this, or any of the legitimate results of what has been called the principle of 'homogeneity.
But the application in this instance was not legitimate, or at all events not legitimately conducted. There was on the face of it an attempt at fallacy, consisting in substituting for the angles the numbers which expressed their ratios. Professor Leslie brought this into full light, by pointing out that if the same reasoning was applied to the case where two sides (a and b) were given and the angle between them P, it would produce the statement that the remaining side c=0: (a, b, P), in which, on substituting for a, b, c, the numbers which express their ratios, there would be the same argument for inferring that c would be the same whatever was the angle, which is notoriously untrue. And this brought out the avowal, that his opponents in the case of the angles intended to substitute the ratios, and in the case of the sides, not; a mode of
les angles A et B, avec le côté p; car, si plusieurs angles C pouvaient correspondre aux trois données A, B, P, il y aurait autant de triangles différents qui auraient un côté égal adjacent à deux angles égaux, ce qui est impossible : donc l'angle C doit être une fonction déterminée des trois quantités A, B, p; ce que j'exprime ainsi, C= 0: (A, B, p).'
• Soit l'angle droit égal à l'unité, alors les angles A, B, C, seront des nombres compris entre o et 2; et puisque C= 0: (A, B, P), je dis que la ligne p ne doit point entrer dans la fonction . En effet, on a vu que Č doit être entièrement déterminé par les seules données A, B, p, sans autre angle ni ligne quelconque, mais la ligne p est hétérogene avec les nombres A, B, C; et si on avait une équation quelconque entre A, B, C, p, on en pourrait tirer la valeur de p en A, B, C; d'où il résulterait que p est égal à un nombre, ce qui est absurde: donc p ne peut entrer dans la fonction P, et on a simplement C=: (A, B).'— Legendre. Elém. de Géom. 12ème édit. Notes. p. 281.
The entire passage is inserted together, to show that no alteration has been made either in order or connexion.
arguing comparable only to the ingenuity of the artist, who in playing at odd or even, holds a ball which he has the power of projecting or not as is required to make him win.
When pushed on this point, they replied, that their reason for substituting the ratios in the case of the angles and not of the sides, was because the right angle was the natural unit of angles*' But the fact of a right angle (or more properly, four right angles or a turning from the place started from till arriving at it again, being a convenient object of reference for the comparison of angles in general, is devoid of any proved connexion with the propriety of substituting the ratios in one case, and not substituting them in the other.
When pressed, however, they produced a reason. They said it was because the angle is a portion of a finite whole, the straight line a portion of an infinite whole; so that every given angle is a finite quantity, while every given straight line is a quantity infinitely small, and only the ratios of given straight lines can enter into our calculations with given anglest. And this was repeated as a very suhtle and very just metaphysical idea ; and at the same time strictly analyticall' On which all that can be done, is to remark on the complete absence of any reasonable or demonstrated connexion (even supposing the terms correct, which might be disputed), between the facts alleged and the consequences assigned to them.
But a circumstance that appears to have escaped Professor Leslie, is that his opponents, till his counter case appeared, had been at the expense of a useless wrong. Whether this arose from mistake, or from foresight of the argument that might be brought against them, might be a curious speculation ; but certain it is, that there was in the first instance no necessity for the substitution, to produce their argument. They would seem to have been beset by the idea, that when the angles A and B appeared on the same side of their equation, one must of necessity be divided by the other ; else why did they insist on the substitution of numbers at all? Whereas the fact they were themselves aiming to prove, was that C=2R - (A + B); where R stands for a right angle. They gained nothing by the substitution of numbers, that they might not have had without; for p would have been just as intractable a companion for the angles as for the numbers. The suspicion consequently may be, that they were lying in wait for Leslie's argument. And when that came, they should have said that they would eject p the side, as incapable of homogeneity,
р but for P the angle they would substitute and then it would
R he a number, which need not be ejected. This would at least have held together ; but it would have sunk under the unreasonableness
L'angle est une quantité que je mesure toujours par son rapport avec l'angle droit, car l'angle droit est l'unité naturelle des angles. Dans cette notion très simple, un angle est toujours un nombre. Il n'en est pas de même des lignes : une ligne ne peut entrer dans le calcul, dans une équation quelconque, qu'avec une autre ligne qui sera prise pour unité, ou qui aura un rapport connu avec la ligne unité.'— Letter of M. Legendre. Leslie's Rudiments of Plane Geometry. Fourth Edition. Notes and Illustrations,
+ Paper of M. le Baron Maurice; as giren in Dr. Brewster's Edition of Legendre's Geometry, Notes, p.235. #Note by M. Legendre, Ibid.
of the substitution demanded in one case with intention to refuse it in the other.
And this leads to the substantial inference from the whole of the somewhat perplexed controversy which took place ; which is, that the original mistake consisted in confounding two sets of things essentially distinct,—the quantities the fixation of which causes an. other quantity to be necessarily fixed or what Euclid in his Book of Data would call given, and the quantities which must be employed as elements in its actual calculation. These two sets are not necessarily alike, either in number or in kind. Take for example Professor Leslie's case, where c=0: (a, b, P). It is quite true that when a, b, P are fixed, c is fixed. But proceed to the actual calculation of and
very different things appear upon the scene. For the yalue of c has to be coilected from the well-known trigonometrical
2R - P formula, that a +bia - 6:: tang. of
: tang. of semi-difference of the angles at the base c. Here then instead of the solitary, and heterogeneous angle P, start up among the practical elements of the calculation two straight lines in the shape of the tangents of two arcs ; which of course do not afterwards fail to conduct themselves with perfect submission to the law of homogeneity. And with all this the proposers of the analytical proof are bound to make their argument square; for the concession of their own demands ends in establishing the results of vulgar trigonometry, and not in altering them. On the whole therefore, the pretence of knowing what quantities must be ejected to preserve the law of homogeneity, is visionary till it is known what quantities may or may not subsequently appear among the practical elements of the calculation, which is impossible in the preliminary stage.
The point, then, which the supporters of the analytical proof must be called on to establish, is why the possibility of the apparition of new elements which is visible in other cases (and which in Professor Leslie's case they actually claim by demanding the admission of R), is non-existent in their own. Take for example the case of what may be called the hyperbolic triangle AHG in p: 140. In this it is plain that if the line AB and the straight lines AG and GH are fixed and determined, the angle AHG must be one fixed angle and no other. But proceed to calculate the comparative magnitude of the angle to different values of AG, and there inmediately start into action new elements in no stinted number, viz. two constant straight lines under the denominations of a major and a minor axis, and a varying straight line under the title of abscissa, to say nothing of the radius of a circle and such sines or tangents of different arcs thereof as may be found necessary in the process.
How then do the opponents know that there are no more elements in the other cuse? If it had pleased nature that the three angles of a triangle should not be always equal to two right angles, the proportionality of the sides of sýnilar triangles would not have lield good, and in making Tables, for example, of the tangents to different arcs of a circle, the magnitude of the radius of ihe circle must in somne grise or other have been an element. The tangent of 45° to a radius of one foot would have borne some given ratio to a fuot, and the tangent of the saine angle. to a radius of two feet, instead of bearing the same ratio to two feet, would have borne some
different one. There must have been a column of numbers to be applied according to the length of the radius, to obtain the true tangent of the angle to a given length of radius ; in the same manner as would be necessary if it was desired to frame a Table for finding the perpendiculars in the hyperbolic triangle for different lengths of base. That this is not so, may be a happy event; but by, what evidence included in their proposed demonstration, do they know that it is not? All they can say is, that they have no evidence that it is so. Their fallacy therefore, is that of putting what they do not know to be, for what they know not to be. Or iť they trust to the difficulty of finding anything in the case of straight lines by which the variation of the angle could have been regulated, - how do they know, for example, that nature instead of making the angle C=2R- (A + B), has not made it=2R-(A + B)x"P, where m the modulus is some given straight linc; m again being equal in
2R different triangles to % X
where % shall be some grand
A + B' modulus existing in nature, which (for the sake of removing the argument from vulgar experience) may be supposed to be of very great dimension, as for instance equal to the radius of the earth's orbit? If an astronomer should arise and declare he had found astronomical evidence that this was true, how would the supporters of the analytical proof proceed to put him down?--and would they not find themselves in the situation of those prophets, who find it easier to prophesy after the fact, than while the result is in abeyance ?*
22. ` A demonstration is offered in the Elémens de Géométrie, par Lacroix (13ème édition, p. 23) and attributed to M. Bertrand, which is perhaps the hardest of all to convince of weakness, and takes the strongest hold of the difficulty which exists in distinguishing between observation and mathematical proof. M. Ber. trand begins by stating, that an angle however small may be multiplied till it equals or exceeds a right angle; whence it may be considered as intercepting a surface equal to or greater than some given fraction of the infinite space intercepted between the straight lines that form a right angle, as for argument's sake a thousandth. If then there be taken another right angle, and at any distance from the angular point be drawn a perpendicular from one of the straight lines that make the right angle and of course a parallel to the other, and other perpendiculars at the same distance in succession from one another; a thousand of the intercepted surfaces or bands, it is argued, will not fill up the infinite space aforesaid; wherefore, it is conçluded, the space between two of the diverging lines must finally be greater than the space between two of the parallel ones, and consequently a straight line making an angle of a thousandth part of a right angle with the straight line to which the first perpendicular was drawn parallel, must finally overtake and pass and cut that perpendicular.
All references to the equality of magnitude of infinite areas, are intrinsically paralogisms. When it is affirmed that the surface of the thousand small
angles is equal to the surface of the right angle and the surface of the thousand bands is not, to reduce this to
• For reference to a number of places where this subject is agitated in various senses, see Legendre's Elém. de Géom. 12ème édition, Notes,
T. C. Hand, Willem 38 ternoster Row. London.
anytning reasonable and precise it is necessary that it be understood to imean, that if a circle be drawn about the angular point, the portion of its area intercepted between M. Bertrand's two perpendiculars to the radius, will, on increasing the radius of the circle while the distance between the perpendiculars remains unaltered, diminish in ratio to the area of the whole quadrant, and may be reduced below the thousandth or other assigned part ; from which it would follow that the line making the angle of a thousandth part of a right angle with the straight line to which the first perpendicular was drawn parallel, will have cut that perpendicular. And there is no doubt that this is accordant with experience in ordinary cases ; but the question is, whether there is evidence that it will of necessity take place in cases where experience cannot be said to act at all. The real basis on which the mind inclines to receive it, is neither more nor less than the confidence felt in the indications derived from experiment in other cases, that the perpendiculars between parallel lines are equal, and consequently the areas of parallelograms vary as their length ; but if this is not to be taken for granted, the question is, what evidence, and of what kind, there is in an extreme case such as may be said to remove itself from experimental examination. Suppose, for example, two perpendiculars to the earth's diameter, and a straight line making with one of them an angle of a second, or 324,000th part of a right angle. In a case like this, where even on taking for granted the theory in dispute the meeting could not take place under a distance equal to above 206,000 diameters of the earth or 17 times the distance of the sun, and in which the difference of the lines from parallelism at any possible portion of their course is totally imperceptible to human sense,-has the mind any convincing evidence, either from experience or otherwise, that it would be contrary to the nature and constitution of straight lines that the two perpendiculars should fail at some time to intercept an area less than the 324,000th part of the quadrant, and by a gradual deflexion totally beyond the perception of the senses should evade the approaches of the would-be secant, and take a form resembling that of a parallel to it, or of a line to which it is an asymptote? Man has experience in some cases that the like would not happen; and it is not denied.that in some cases the section will take place. But has he experience in this? and if he has not, has he geometrical evidence to put in its room? It is imagined, not. This then may be concluded to be another, though a very complicated and ingenious case, in which man's empirica. knowledge of what he has tried, is substituted in what he has not tried and cannot try. Which, whatever may be thought of the chances of its final validity, is not geometrical proof.
The number of demonstrations proposed on the subject of Parallel Lines is evidence of the anxiety felt by geometrical writers upon the subject. If an erroneous account has been given of any cited ahove, provision has been made for furnishing the reader with the references required to put him in possession of the truth.