« PreviousContinue »
we should see, in like manner, how in this case the proposed equation may be reduced to a relation between abstract numbers.
What has now been said is enough, without doubt, completely to illustrate the question. We may add, however, that if we chose to ascend to a more elevated analytical principle, it were easy to shew, that between angles and one straight line, there cannot exist any relation from which the latter, as a function of the former, can be determined. These quantities considered as magnitudes destined to enter into our calculations, are not homogeneous, when referred to the wholes of which they respectively form part. 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, whilst every given straight line is a quantity infinitely small, and only the ratios of given straight lines can enter into our calcu lations with given angles."
It is upon this principle only that we can explain the analytical paradox which seems to result from the relation C (A, B, c) not being capable of admitting the solution c (A, B, C). By the foregoing considerations, it is evident that this solution is impossible, and the geometrical indetermination of the question corresponds to this analytical impossibility.
The objections stated above may now be considered as entirely dissipated; a few reflections may, however, be subjoined.
Mr Legendre is reminded, that heterogeneous quantities are considered simultaneously in the science of Mechanics. But every one knows, that in this science, all equations expressing any relation between space and time, will, besides these two elements, of necessity include a line-unit and a time-unit. The equation considered, therefore, will not be sim
ply between s, s'
S and T being the two units of space and time; so that
no equation of this kind will ever exist, except between abstract numbers. And if velocities also are supposed to enter into this equation, is it not, in like manner, well known, that these velocities, mere conceptions of our own mind, are nothing but abstract numbers resulting from
Another objection is sought for in the dependence which exists between an arc of a circle, its radius and the angle to which it corre
* This is a very subtle and very just metaphysical idea: it is, at the same time, strictly analytical; and explains why c stood alone in the equation C (A, B, c), where c may be suppressed, because it is infinitely little, or nothing. It also explains why, in the case where two lines a and b are found in an equation containing no other quantities but numbers; these two lines cannot exist in the equation except by their ratio so that making b
would of necessity disappear from the equation.
ap, the quantity, a
Note by M. Legendre.
sponds. But a logical relation, such as our mind shews us to exist between these three quantities, differs essentially from the actual determination of one of them, by means of the other two; in other words, from the rigorous and mathematical expression of the former in terms of the latter. And, indeed, when the length of a circular arc is expressed in parts of its radius, is there any trace of an angular quantity in its expression? Nothing more is done than compare approximately the length of a certain circular arc, described by means of a given radius, with the length of this radius itself; and as for the angle to which it corresponds, it disappears entirely from this operation.
In the last place, after having very justly observed that the measurement of angles depends on the subdivision of a whole revolution performed round a point in a plane, the author of the objections (and we cannot help feeling astonished at his doing so), abruptly confounds two things which are essentially distinct; namely, the determinate ratio of a given angle to this total angular quantity which nature presents to us,-with the quite indeterminate ratio of a given straight line, taken as the standard of measure, to the straight line in general,-which being infinite in its nature, and therefore never given or assignable, there can exist no ratio whatever in the case. Here there is no room for analogy. Whilst our angles are quantities constant of themselves, all our lines, like spaces and times, are nothing but ratios between two arbitrary quantities; the metre, for example, and the radius of the earth; or, in astronomy, this radius and the distance between the centre of our planet and the centre of the moon or of the sun.
Let us finally examine the objections of this anonymous geometer, who, in Mr Leslie's opinion, stands decidedly at the head of British mathematicians. We shall state them faithfully, like the foregoing.
"With regard to Legendre's demonstration, I am of opinion, that there is involved in the mise en equation (reduced to an equation), a principle which is equivalent to Euclid's 12th axiom, (if a straight line meets two straight lines, so as to make the two interior angles on the same side of it together less than two right angles, these straight lines being continually produced, will at length meet on that side on which are the angles which are less than two right angles). Using the notation of your book, his assumption is, that C = 4: (A, B, c). Now this means, that we shall get the angle C, by combining the angles A and B with the line c, in a certain way; and it is implied that this is true, whatever value the line c have; or, in other words, it is true for all values of c. Suppose, then, an individual triangle, of which c is the base, and A, B the angles at its extremities; conceive an indefinite number of lines, of any lengths, c', c", c'", &c. and at the ends of each of these lines, angles to be made equal to A and B ; will a triangle be thus formed upon each of the lines c', c', c'", &c. or not? If you say that you cannot allow the existence of such triangles without proof, you agree with the Greek geometer; but then you must deny the legitimacy of Legendre's equation C = ?: (A, B, c); for it supposes the possibility of such triangles, since it is a determination of the third angle of each of them, from knowing the base and the other two angles. If you grant the possibility of the triangles, then Legendre's equation will be established; but you also admit Euclid's 12th axiom. For
you assume, that two lines drawn at the extremities of any third line, so as to make with it two angles equal to any two angles of a triangle, do meet one another when produced. On examination, you will find that the only relation generally true of two angles of a triangle is this, that they are together less than two right angles. I cannot, therefore, admit, that Legendre's demonstration contributes in any degree to remove the difficulty in geometry. The intrinsic evidence of a principle or proposition, is the same whether it be expressed in common language or translated into the language of functions. Grant to the geometer the same assumption which is implied in the functional equation of the analyst, and he will be no longer embarrassed with the theory of parallel lines. Legendre endeavours to justify his equation, by saying, that two triangles are identical when they have their bases equal, and likewise the angles adjacent to their bases equal, each to each. But this does not prove, that of all the infinite number of triangles which can be formed upon a line greater or less than the base of a given triangle, there is always one that has the angles at its base equal to the angles at the base of the given triangle. If this be thought a more self-evident principle than those that geometers have employed, let it be transferred to geometry, and that science will no longer have need to borrow aid from the theory of functions."-Leslie's Geometry, pp. 294, 295.
This argument, it must be allowed, has something in it new and ingenious; yet, at bottom, the comparison on which it is founded, the identity attempted to be established between M. Legendre's equation and the famous postulate of Euclid, is only specious, not at all solid. In the first place, if it were not true that a triangle may always be considered as sufficiently determined by one of its sides and two adjacent angles, what would become of Mr Leslie's demonstration of his own twentieth Proposition? What would become of this Proposition itself, which establishes the possibility in question, and which has been regarded by all geometers as certain, and as an evident consequence of the principle of superposition? But, farther, the point of view to which it has pleased this ingenious correspondent to transport himself, in the comparison which he attempts to institute with Euclid's alleged axiom, is altogether different from that in which M. Legendre has wisely placed himself. Just as it is mathematically allowable to doubt respecting the actual meeting of two straight lines, originally perpendicular to a third, when one of these lines has ceased to be perpendicular to the latter; just so, it is equally impossible to raise any doubt with regard to this meeting, when we suppose, as of necessity is done here, that the triangle in question actually exists. And how, indeed, could we engage in the consideration of any property of triangles, if these triangles were not before-hand looked upon as formed, and, consequently, as possible? We are not here inquiring, whether, under certain conditions, the triangle will exist; but when it does exist, we wish to know what conditions are required for determining it; and these being once known, and given by the principle of superposition, we may reason with absolute strictness from the supposition we have made, whatever be the magnitude of the sides and angles of this triangle.
Accordingly, what happens to this anonymous geometer? In the path he has chosen to follow, after having thus distorted the condition of
the question, he meets, and quite gratuitously, with all the difficulties inherent in the direct establishment of the theory of parallel lines, and arising from the idea of infinity, which is immediately connected with that of the straight line in general; whilst M. Legendre, naturally setting out from the existing triangle (and we have seen that this proceeding cannot be objected to without completely changing the state of the question), finds means to deduce this very theory from the proposition concerning the sum of the three angles. He thus most skilfully eludes the difficulty, by taking advantage of the indisputable equation C=4 (A, B, c) in order to get rid of every idea of infinity by means of eliminating the straight line c; an operation depending, as we have seen, on a principle as rigorously true as any of those which have been hitherto admitted in geometry.
The force of M. Legendre's demonstration is therefore derived from the fact, that a necessary law of the calculus, when applied to any relation indicated by geometry, is to retain in this relation nothing but quantities of a comparable magnitude; and the art of its author, on this occasion, has been to put under the decision of the calculus which implied the necessity of such an elimination, the solution of a question which geometrical considerations alone left in its primitive complexity. In fine, if we make the superfluous remark, that the same mode of reasoning, when employed to demonstrate several other very important propositions, in the remaining part of M. Legendre's second Note, has always guided its author to correct results, whilst no one has succeeded in legitimately deducing from it a single consequence in itself incorrect, we shall be warranted in concluding from all that precedes, that the demonstration attacked in Mr Leslie's work is perfectly rigorous. algorithm of functions had previously been employed with success in establishing the fundamental principles of Mechanics (see the Miscell. Taurin. tom. I. and II.); but this new application of it does not yield to those which have gone before; indeed, such a result was to be expected from so happily introducing into the fundamental relation which geometry furnishes, a principle so evident and incontestable as that of homogeneity. F. M
On the Approximation employed in Prop. 16. IV.
So soon as we have found a radius too great and a radius too little which agree in their first ciphers, the calculation may be completed in a very speedy manner by means of an algebraical formula.
Let a be the defective radius, and b the excessive one, their difference being small; let a' and b' be the radii next in order deduced by the What we are in quest of is the
formulas b'ab, a'=√(a.
last term of the series a, a', a", &c., which at the same time will be the
last of the series b, b', b", &c. Let this last term be named x, and put b=a (1+w); we can suppose xa (1+Pw+Qw2 + &c.), P and Q being indeterminate coefficients. Now the values of b' and a' give b'=a (1+}w — } w2 + &c.) ;
a'=a (1+1 w2 — z zw2 + &c.)
And if, in like manner, we put b' = a′ (1+w'), we shall have
But the value of x must be the same whether the series a, a', a", &c. begins with a or a'; hence we shall have
a (1+Pw+Qw2 + &c.) = a' (1+Pw2+Qw22 + &c.) Substitute, in this equation, the values of a' and ' in terms of a and w; and comparing the similar terms, we shall deduce from it P = 3, and == -1; hence
x = a (1+} ~ — 1 z w2).
If the radii a and b agree in the first half of their ciphers, the term w2 may be rejected in the calculation, and the preceding value will be
reduced to xa (1+w)=a+
Thus making a=1.1282657,
and b = 1.1286063, we shall immediately deduce from it x= 1.1283792.
If the radii a and b agree only in the first third of their ciphers, we shall have to take in the three terms of the preceding formula ; and thus making a = 1. 1265639, and b = 1.1320149, we shall find x=1.1283791.
We might suppose a and b to be still farther different; but in that case it would be requisite to calculate the value of x with a greater number of terms.
The approximation exhibited in Prop. 14, the author of which is James Gregory, is susceptible of similar abridgments. We refer to Gregory's book, entitled Vera Circuli et Hyperbola Quadratura, a work of great merit, considering the time when it appeared.
Shewing that the Ratio of the Circumference to the Diameter, and also its square, are irrational Numbers.