Hence a=a+(b+e),. where v is any finite cardinal number. If now, we can conclude, from (7), that also a=a+(b+c), we can say, because the right-hand side of (7) becomes 1+) +.א(e=[a+(b+e)]+e, (7) (8) which is the equation required. This part of the proof is purely cardinal; the ordinal part appears in the proof of equation (8) from equations (7); in other words, in the proof that it is possible to conclude from {} to in the case of (7). From the manifold M1, of cardinal number a, we can take away, by (6), a manifold (P1) of cardinal number while the remainder has still the cardinal number &; let M2 be this remaining manifold. From M2 we can again (by (7)) take in a similar manner, a manifold (P2) of the cardinal number (10), and we thus obtain another remainder M3 of + That one cannot, in general, conclude from {v} to is evident from the consideration of the known relations: אאאאא"א That, however, this "extended principle of cardinal induction" is legitimate in the case of the text, I had found, independently of Zermelo, in October, 1902; and the remark of the ordinal character of the proof has led me to emphasize this point in Zermelo's proof. The extended principle of ordinal induction, or conclusion from {v} to w, as used by Schönflies (op. cit. pp. 45, 52, 60, 67, 235), should be compared with this. It seems true that mathematics is principally occupied with sufficient (and necessary and sufficient, in closer investigations) conditions under which one can conclude from {v} to w. Thus, if {8,(x)} be a convergent sequence of functions, and sw(x) thus properly denotes its limit, the most fundamental problem here is to know when one can conclude from {s(x)} to s(x) as to continuity, regularity, etc., and uniformity of convergence is important because it gives a wide sufficient condition. cardinal number a. Proceeding in this way, we obtain a series of manifolds P2, P3, P4, ..., Р,, ..., (11) and this series can stop at no finite v, for, if it did, the equations (7) would be contradicted. Further, each P, has no point in common with any other Pu and there exists † a manifold μ which may, however, consist of no elements, which is the first (in the above process) of all manifolds which is not contained in all the manifolds P,. This Mw is thus defined by essentially ordinal considerations. Let, then, g be the cardinal number of Mω ; then, since the series (11) is of type w, and therefore of cardinal number א, while each P, is of cardinal number (10); we have a=x(b+c)+g =2 .א)b+c)+g=]א)b+c) + g[ + א)b+e( which is the required equation (8). Accordingly, the independent proof of the theorem B appears to depend essentially on ordinal conceptions, although it is true that whatever may be the cardinal numbers of M and N, the proof requires only an enumerable manifold (of type w + 1) of steps. 9. We may now deduce some general laws of calculation with transfinite cardinal numbers, analogous to those given by Cantor § for א; namely, where v is any finite cardinal number, For this purpose I shall now prove the first of the two theorems denoted by Whitehead || in his memoir “On Cardinal Numbers" as unsolved, namely: If a and b are cardinal numbers, a is transfinite, and + Cf. Schönflies, op. cit. p. 14. † Cf. Schönflies, op. cit. p. 18. § Math. Ann. xlvi. pp. 492-495. Amer. Journ. of Math. xxiv. pp. 367-394 (1902); see especially, pp. 368, 381-383, 393. If a belongs to the class considered by Whitehead † in a series of interesting propositions, and which is characterized by the property that or, what is evidently the same thing, that there exists a cardinal number such that Also-and this was not pointed out by Whitehead-from (13) follows (12). For we can apply the conclusion from {v} to of to the equality a=a+a=v. A, which results from the hypothesis. (14) If, now, Mis a definite one of the well-ordered manifolds of cardinal number a, and we replace each element of M by a well-ordered manifold of two elements; then, since M consists of a cardinal number b‡ of series, each of which is of type w, together with perhaps, a finite number (v) of elements §, the manifold resulting from M also consists of b series, each of which is of type w, together with, perhaps, a finite number (2v) of elements. This results from the known equation 2.ω=ω. Since then, the resulting manifold is of cardinal number we get equation (14). a+a, Thus if & is any transfinite cardinal number, a is unaltered by the addition of any (finite or transfinite) cardinal number bequal to or less than a, and of these alone. Accordingly the class of such numbers b is hereby completely determined, and consequently for any cardinal number a the following rules of calculation hold: + Op. cit. p. 393. It is easy to see that b=a. Thus, i = M may be of type 2+w+v. The first two equations contain the properties proved by Zermelo for the class of numbers b, which, however, was not determined by him. This class of numbers was called by him a "group belonging to a" inasmuch as the members reproduce themselves or other members of the class by their diminution, by their multiplication with א, and by their addition in a finite or enumerable manifold of summands. If we add that the members also reproduce other members of the class also by increase, when this is necessary, till they become equal to a, we have a characterization of the group in question. Further, we shall prove later that which gives a further self-reproductive property of members of the group. 10. If we attempt to use the method of § 9 to prove the second theorem denoted by Whitehead † as unproved; namely, or its equivalent a.b=a (ba), a2= a; (15) we are met by the fact that, just as it is necessary to have proved that in order to prove ‡ that א=א א <א وں so it is necessary to prove previously the equation (15), or אא where y is any ordinal number, before the existence of the series of Alephs (1) can be proved. It is, then, necessary to investigate in greater detail the series (1) ‡. The importance of this may be considered as established by the fact that, having arrived at (1), we are sure that every transfinite † Op. cit. p. 368, Whitehead remarked that (15) does not follow from (12). I See 'Grundlagen,' pp. 35-36; Math. Ann. xlix. pp. 227-228, 222. § Cantor has hitherto only treated in detail the ordinal number of the first two classes and the cardinal numbers of these classes. cardinal number occurs in it; and, further, we shall, by this investigation, obtain a complete solution to the problems of determining: (1) The result of adding any (finite or transfinite) number of any cardinal numbers; (2) The result of multiplying any finite number of any cardinal numbers. There will only then remain the consideration of those cardinal numbers of the form ab where bis transfinite. Some results as to this class of numbers, together with the detailed investigation of Cantor's "number-classes” in general mentioned above, I will give in a continuation of this paper. Little Close, Yateley, Hants. VIII. Note on Borgnet's Method of Dividing an Angle in an Arbitrary Ratio. By Prof. J. D. EVERETT, F.R.S.* I HAVE recently come across an old paper (Borgnet, in Rouen Acad. Travaux, 1839, pp. 113-143) containing a beautiful theorem which seems to have fallen into oblivion. The paper is devoted to what the author calls barycentrides, a barycentride being defined as the locus of the centroid of an are (of any curve) measured from a definite initial point. The theorem to which I refer solves, by means of the barycentride of a circle, the general problem to divide a given angle in the ratio of any two given straight lines. In fig. 1 let P be the centroid of the circular arc AQ, and let the curve AP be the barycentride of the circle AQ described about O. Bisect the chord OP at right angles by MH, meeting in H the perpendicular at O to the initial radius OA. Let 0 denote any one of the three equal angles AOP, OHM, MHP; then we have OP=20H sin 0. But by the rule for the centroid of a circular arc * Communicated by the Author. |