HATIOS, VKl.MK AND ULTIMATE. 



HATIOS, I'lilMi: AND IM.TIMATE. 



til 



but th* bet efforts against it were nude by the Supernaturalistic 

 choc! Rationalism in ita first form seems to hare received it* death- 

 Wow from thU work anil the various controversial writings of Strauss 

 and other*, while the new school made rapid progress. The con- 

 tort between it and the Supernaturalista was however, with i 

 ccptioos, conducted in a calm and purely philosophic manner. l'--tli 

 parties were honest enough to give way whenever any of tln-ir disputed 

 point* were proved to be untenable. "i.-mifi-st from the 



Life of Christ ' published by Ncandrr, and from the different editions 

 of the work of Strauss. 



Notwithstanding the wide breach between tho two parties, approxi- 

 mations were made on both sides, so that the Supematuralists as well 

 as the Rationalists might each be divided into two parties. Some of the 

 latter, such as Paulus, Wegscheider, CJesenius, Schulthess, Schulz, nnd 

 others, continued to consider reason as their only guide in matters of 

 religion, and rejected every supernatural revelation : others, the so- 

 called Supernaturalistic Rationalists, admitted indeed a supernatural 

 revelation, but considered reason as the only means of recognising and 

 acknowledging it; they thus still allowed reason to be the supn me 

 judge in matters of religion. To this class of Rationalists belong 

 Bretschneider, Von Ammon, Bohme, Hose, Roster, and others. A 

 similar division exists among the BoperDCtanlMt, 



The view which the Rationalists had taken of the Scriptures con- 

 tained some elements which led to a new crisis in German theology. 

 Some parts of the Scriptures, from which the Rationalists, with all 

 their ingenuity, saw no hope of eliciting a genuine history, they 

 had ventured to declare to be a mere legend, tradition, or niytlius. 

 This view was gradually applied to a great portion of the Old Testa- 

 ment, as hi Bauer's Hebriiische Mythologie,' Leipzig, 1802. The 

 various and profound investigations into ancient profane history had 

 led to similar results in other departments, and the hollow-ness of the 

 Rationalist interpretation was either loudly proclaimed or tacitly 

 acknowledged by all parties. The consequence was either a return to 

 the Supernaturalistic view, or further progress in the path which had 

 >>een opened by the nationalists themselves. Those Rationalist! who 

 could not do the former now applied the principle, to which they had 

 formerly recourse only in cases of extreme difficulty, to the whole body 

 of the early and miraculous portions of the Scriptures, which they 

 placed on the same footing with the early and fabulous stories ol 

 ancient Greece and Rome, and considered as a mythical history not 

 written by eye-witnesses or contemporaries, and only recorded after it 

 had been handed down by tradition through many generations, 

 i ding to this view, all the events in the Bible are either natural 

 i, such as occur in the history of other nations, and which must 

 be examined according to the general principles of historical criticism 

 or they are of a miraculous and supernatural character, and must for 

 this reason be rejected as not historical, like the fabulous accounts 

 of ancient mythology. As the Rationalistic school directed its first 

 attacks against the deists, so the mythical school, though diametrically 

 opposed to the Supernaturalista, directed its main efforts against 

 Rationalism. We must nevertheless consider this last school as essen 

 tially Rationalistic, or as a second form of Rationalism, in as far as 

 like Rationalism in ita first form, it takes reason for its sole guide, anc 

 denies all supernatural revelation. The only difference is that it denies 

 the Biblical records to be the works of eye-witnesses and contempo 

 lanes, and hence draws the conclusion that it is utterly impossible to 

 (licit from those portions which are supposed to consist of mythica 

 stories anything like a true and connected history. 



Rationalism in the German sense never made much way in England 

 being chiefly confined to a portion of the Unitarian sect. But of late 

 years more recent principles of historical and philological evidences 

 with the incontestable facts established by geological and pahconto 

 logical investigation, shave led to closer though more reVerent approache 

 to it from various quarters. The most remarkable is what is known 

 as the new school of divines at Oxford, of whose views the fullcs 

 development is afforded by the works of the late Baden Powell and tin 

 Rev. B. Jowett ; and the ' Easavx and Reviews ' issued by these gentle 

 mcr, and the Rev. J. Temple, Dr. Williams, Rev. Mr. Wilson, Mr 

 Goodwin, and Rev. Mr. Pattison, in 1860. The more orthodox view o 

 the question has not been wanting of numerous supporters. 



Details, of the various opinions of many of the writers mentioned in 

 this article will be found under their respective names in the Bioo. Div., 

 and under MIRACLES there is given a list of foreign writers, nearly all 

 of whom are purely rationalistic. 



RATIOS, 1'RIMK AM) ULTIMATE. These terms were first 

 introduced, at least in a system, by Newton, who preferred them to 

 the terms suggested by his own method of fluxions. The first section 

 of the 1'rincipia contains the development of their meaning, with 

 various propositions enunciated in their language. In the articles 

 LIMIT and IM I.VITE we have already had the same notions to consider, 

 couched in different words ; but when we remember that the only sure 

 f'Mind.ition ,,f tin- differential calculus, that is, of all the higher part of 

 mathciii.-itiri. mint rest upon these notions, it will be worth while to 

 dwell n little UIHUI Newton's form of expression, and his method of 

 emploj'infj it. The notions in question actually form part of the 

 knowledge of rn;iny pi TSOIIS who are not mathematicians, and all whoso 



lead them to any considerations connected with > 

 mi-lit, must in some degree possess and appeal to them. 



All who understand the term ratio must see that the ratio of two 

 (U.intities doe* not depend on their actual magnitudes. If one 1 

 o another in the ratio of 3 to 7, the halves, thirds, fourths, &c. of :h> 

 wo lines will have the same ratio; and the subdivision into ;. 



, iy bo continued without limit : thus the hundred-millionth 

 art of one line will be to the hundred-millionth part of the other as 3 

 o 7. Ratio then always exists, so long as there is magnitude ; but if 

 nngnitude should cease to exist, .m.l if both lines should vanish, no 

 dea of ratio can be formed. If however the diminution take place by 

 continual subdivision, this evanescence of magnitude never takes place ; 

 'or into how many parts soever a line may be divided, each part is a 

 length, still subdivisible for ever. 



The consideration here introduced is not an easy one at first, for then- 

 is a degree of smallness which evades the senses, and reason must come 

 to their assistance. This makes a great difficulty, for many who think 

 themselves rational geometers are not aware how much of their 

 ordinary perception of geometrical truth is the consequence of what 

 they see, not of what they deduce. All magnitude is relative, 

 as the notion of great or small is connected with it ; we know thia 

 when we stop to think, but we do not easily take it along with us in 

 our thoughts ; there is nothing absolutely great or small, but we are 

 continually making an absolute greatness out of magnitude which i-. 

 grent compared with our own bodies, and an absolute smalhicss of 

 that which is in the same sense comparatively small. 



Take A o, an arc of a circle, A,o its half, A 3 o its third part, A,o its 

 fourth part, and so on; let the chords AO, A a o, A 3 o, &c. be drawn. 

 The points A,, A 2 , A s , A,, &c., constitute a series continually approach- 

 ing to o in position, but never reaching it, for no aliquot part of A n 

 is absolutely nothing. Now it can be shown that A , o, the chord of 

 (A , o) the .rth part of the arc, will be nearer to a ratio of equality with 

 (A , o) the greater x is taken, so that any approach to equality may be 

 attained and passed by making ,r sufficiently great. The beginner's 

 notion is, almost invariably, that two small quantities must be nearly 

 equal, because they are small ; and the fallacy under which they 

 proceed is the following: quantities which are nearly equal to the 

 same are nearly equal to one another ; small quantities are all nearly 

 equal to nothing, therefore small quantities are nearly equal ( 

 another. The mistake here lies partly in the use of nothing as if it 

 were a quantity, having all the properties of quantity, jwrtly in the 

 supposition that quantities which differ little must be nearly equal. 1 t 

 by differing little be meant that the difference is trifling when com- 

 pared with the quantities themselves, the notion is a good one : two 

 microscopic animalcules arc nearly equal when they differ by a small 

 jx>rtion of an animalcule ; but if they differ by the size of a gnat, 

 though their absolute difference is still small, compared with our usual 

 standards, the larger is immensely greater than the other. But if the 

 just notion of nearly equal be adopted, it is wrong to say that the 

 chord and arc are nearly equal on account of their smallness, since their 

 small difference may possibly itself be larger than one of them. And 

 as to using nothing as a quantity in the fallacious syllogism above 

 given, it must be remembered that, with reference to possibility of 

 subdivision, any quantity, however small, is as distant from nothing as 

 any other quantity, however great, is from infinity. 



Nevertheless, as may be rationally shown, the chord and arc are the 

 more nearly equal the smaller they are. The conception of tin 

 position may be aided as follows: however small a line may be. we 

 may represent it by as great a number as we please, if we take the 

 unit of measurement still smaller, and sufficiently smaller ; now let the 

 arc taken be the nth part of the radius; then if a unit be ti! 

 small that the arc shall be represented by 24n*, the chord will be a 

 fractional number extremely near to 24n 3 1. Thus if the arc be 

 one-thousandth of the radius, and a unit be taken to measure it which 

 is its 24-thousandth part, so that the arc is _' I.IHIII, the chord will 

 contain that unit a very small fraction more than _':!. I'!'!* times. And 

 if w be made still greater, the inequality will be made still less, being 

 capable of being made less than a unit out of any number we may 

 name, however great. 



In the article LIMIT we should Bay that the limiting ratio of tin- ,-n e 

 and chord is unity ; in INTIMTF.. that an infinitely .small arc is e T i.-d to 

 its chord. Newton's phrase was that the arc and chord are ultimately 

 equal, or that their ultimate ratio is one of equality. He strives to 

 his language as much as possible in the Scholium which termi- 

 nates the first section, and from which we now ( ]uoto. 



" I 1' i-d these lemmas, that 1 might avoid the tedium of 



long d> .1. with reductions ad a 



irnts. Demonstrations are shortened indeed by the Method of 

 indrvi.-il.lv..." [CAVAXIBRI, in Bioo. Inv.j "JJutM'i 

 is somewhat ditficult, and the method is not thought very geometrical, 

 I liave preferred to make what follows depend upon the ultimate sums 



and rat Ung ijuantiticK 1 do not wish to be iinder- 



i- using indivisibles, but divisible vanishing quantities: n-.t 



