890 



SCIENCE 



[N. S. Vol. XXXVII. No. 963 



perfectly evident that they do, and with 

 this we might be content. It will be worth 

 while, however, to examine the matter a 

 little more attentively. Denote by L any 

 chosen line or ray of the sheaf. Choose 

 any convenient unit of length, say a mile. 

 We now ask: how many of our units, how 

 many miles can we, starting from P, lay 

 •ff along L 1 Lay off, I mean, not in fact, 

 but in thought. In other words : how many 

 steps, each a mile long, can we, in traver- 

 sing L, take in thought 1 Hereafter let the 

 phrase "in thought" be understood. Can 

 the question be answered? It can. Can it 

 be answered definitely? Absolutely so. 

 How? As follows. Before proceeding, 

 however, let me beg of you not to hesitate 

 or shy if certain familiar ideas seem to get 

 submitted to the logical process — the mind- 

 expanding process — of generalization. 

 There is to be no resort to any kind of 

 legerdemain. Let us be willing to tran- 

 scend imagination, and, without faltering, 

 to follow thought, for thought, free as the 

 spirit of creation, owns no bar save that 

 of inconsistence or self-contradiction. Con- 

 sider the sequence of cardinal numbers, 



(5) 1, 2, 3, 4, 5, 6, 7, ... . 

 The sequence is neither so dry nor so 

 harmless as it seems. It has a beginning; 

 but it has no end, for, by the law of its 

 formation, after each term there is a next. 

 The difference between a sequence that 

 stops somewhere and one that has no end 

 is awful. No one, unless spiritually un- 

 born or dead, can contemplate that gulf 

 without emotions that take hold of the 

 infinite and everlasting. Let us compare 

 the sequence with the ray L of our sheaf. 

 Choose in (8) any number n, however 

 large. Can we go from P along L that 

 number n of miles? "We are certain that 

 we can. Suppose the trip made, a mile 

 post set up and on it painted the number 

 n to tell how far the post is from P. As n 



is any number in (S), we may as well sup- 

 pose, indeed we have already implicitly 

 supposed, mile posts, duly distributed and 

 marked, to have been set up along L to 

 match each and every number in the se- 

 quence. Have we thus set up all the mile 

 posts that L allows? We are certain that 

 we have, for, if we go out from P along L 

 any possible but definite number of miles, 

 we are perfectly certain that that number 

 is a number in the sequence, and that ac- 

 cordingly the journey did but take us to a 

 post set up before. What is the upshot? 

 It is that L admits of precisely as many 

 mile posts as there are cardinal numbers, 

 neither more nor less. How long is LI 

 The answer is : i is exactly as many miles 

 long as there are integers or terms in the 

 sequence {S). Can we say of any other 

 line or ray L' of the sheaf what we have 

 said of L? We are certain that we can. 

 Indeed we have said it, for L was awl/ line 

 of the sheaf. May we, then, say that any 

 two lines, L and L', of the sheaf are equal? 

 We may and we must. For, just as we 

 have established a one-to-one correspond- 

 ence between the mile posts of L and the 

 terms of ((S), so we may establish a one-to- 

 one correspondence between the mile posts 

 of L and those of L', and what we mean by 

 the equality of two classes of things is pre- 

 cisely the possibility of thus setting up a 

 one-to-one correlation between them. Ac- 

 cordingly, all the lines or rays of our sheaf 

 are equal. We can not fail to note that 

 thus there is forming in our minds the con- 

 ception of a sphere, centered at P, larger, 

 however, than any sphere of slate or wood 

 or marble — a sphere, if it be a sphere, 

 whose radii are the rays of our sheaf. Is 

 not the thing, however, too vast to be a 

 sphere ? Obviously yes, if the lines or rays 

 of the sheaf have a length that is indefi- 

 nite, unassignable; obviously no, if their 

 length be assignable and definite. We have 



