RATIOS, PRIME AND ULTIMATE. There can be little doubt that Newton dis covered. by means of flexions, of which lie was in possession at a very early age, the greater part of that extraordinary series of theorems regarding motion, etc., which he first published in the Principle. He bad, however, a great partiality for the synthetic form of demonstration, employed with such success by the Greek geometers; and the consequence was that, in the Principle, he avoided entirely the use of analysis by fluxions, and invented for synthetical applications the closely allied method of prime and ultimate ratios. The fundamental idea involved in fluxions, prime and ultimate ratios, and the differential calculus, is the same, that of a limit (q.v.).
To give an idea of the nature, as well as to show the origin of the name, of the method, we may take a very simple case. Let a particle be projected in the direction AP; it will move uniformly in that line forever, unless deflected from it by some external force. See MOTION, LAWS OF. Suppose that gravity alone acts upon it, then (see PROJECTILES) it will describe a parabolic path, AQ, to which AP is the tangent at A; and the line PQ, which joins the disturbed and undisturbed positions of the particle at any instant, is vertical. Now, the lengths of AP and AQ are not, in general, equal, but they are more and more nearly equal as both are smaller; and, by taking each enough, we may make the percentage of difference between them as small as we choose. In other words, their prime ratio, just at A, is unity. Again, the inscribed square is less than a circle; the octagon greater than the square, but less than the circle; the regular polygon of 16 sides greater than the octagon, but less than the circle; and so on, constantly doubling the number of sides. But it can be shown that the difference of area between the polygon and the circle may be made as small a percentage of the area of the circle as we please, by the sides of the polygon numerous enough. Hence, the ultimate ratio of the areas of the circle, and inscribed polygon with an indefinitely great number of equal sides, is unity.
The basis of the method, which is implicitly involved in the foregoing illustrations, is Newton's first lemma : and the ratios of quantities, which tend con stantly to equality, and may be made to approximate to each other by less than any assignable difference, become ultimately equal." In other words, if we can make the percentage of difference of two quanti ties as small as we choose, we must produce ultimate equality.
From this, in his second and third lemmas, Newton proves the fundamental principle of the integral calculus as applied to the determination of the areas of curves, by showing that if a set of parallelograms, as in the figure, be inscribed in any curvilinear space, the percentage of difference between the sum of their areas and that of the curve may be made as small as we please by diminishing indefinitely the breadth of each parallelo gram, and ineyeasing their number proportionally.
Next, he shows how to compare two curvilinear spaces, by supposing them filled with such parallelograms, each of the first bearing to one of the second a constant ratio.
Next, that the homologous sides of similar curvilinear figures are proportional,.
The sixth lemma is merely a definition of continuous curvature in a curve, as distin guished from abrupt change of direction.
The seventh, eighth, and ninth lemmas are of very great importance. The general principle involved in their proof is this—to examine what occurs in indefinitely small arcs, by drawing a magnified representa tion of them such as always to be on a finite scale, however small the arcs themselves may he. Thus, to show that the chord of a small arc is ultimately equal to the arc—of which we have in trigonometry (q.v.) as a particular case, the ultimate equality of an are and its sine—he proceeds somewhat as follows: Let AB be an arc of continued curva ture, AC the tangent at A. Produce the chord AB till it has a finite length, Ab.
Describe on Ab, as chord, an arc similar to AB. This, by a previous lemma, will touch AC at A.. Now, as B moves up to A, let the same construction be perpetually made, then b will approximate more and more closely to AC (because the arc AB is one of continuous curvature), and the magnified arc will constantly lie between AC and Ab. Hence, ultimately, when Ab and AC coin cide in direction, the arc Ab (which is always between them) will coincide with Ab. Similarly, AD being any line making a finite angle with AC, draw DBE cutting off a finite length from AD; this process enables us to prove that the triangles AED, and the rectilinear and curvilinear triangles ABD, are all ultimately equal.
Finally (and this is the step of the greatest importance in the dynamical applica .
Lions). if the lines AD, DE, D'E' be drawn under the above restrictions, the ultimate ratio of the curvtlutcar of reenhuear triangle AEB, is that of the squares of correspond sides. From this, in the ninth and last lemma, it is easily shown that the spaces de scribed under the action of a finite force have their prime ratios as the squares of the times: whence we pass at once to the ever-memorable investigations of the l'rincipia regarding the orbits described under the action of various forces.
The method of prime and ultimate ratios is little used now (except in Cambridge, which noes 1101101' to itself nt making part in me cipia a subject of study), as the differential and integral calculus help us to the required results with far greater ease. But to the true student of natural philosophy, the synthetic method of Newton is of very great value, as it shows him clearly at every step the nature of the process he is carrying out, which is too apt to be lost sight of entirely in the semi-meehanical procedures common to all forms of symbolical reasoning.