November 14, 1S84.] 



SCIENCE. 



453 



and, it may be said, the oviparity of the monotremes 

 firmly established, the fact had been authoritatively 

 proclaimed. Sir John Jamison, for instance, espe- 

 cially declared that ' the female is oviparous, and lives 

 in burrows in the ground ' ( Trans. Linn. soc. London, 

 xii. p. 5S5). The Rev. Dr. Fleming, in his 'Philoso- 

 phy of zoology' (ii. 215), published in 1822, remarked, 

 that, "if these animals are oviparous (and we can 

 scarcely entertain a doubt on the subject, as the eggs 

 have been transmitted to London), it would be interest- 

 ing to know the manner of incubation." Further, 

 Fleming refused to admit the monotremes among 

 the mammals, dividing the Vertebrata ' with warm 

 blood ' into ' quadrupeds ' and ' birds,' and the former 

 into 'I. Mammalia' ('1. Placentaria' pedota and 

 apoda, and '2. Marsupialia'), and ' II. Monotremata.' 

 But, notwithstanding all these facts, scepticism as 

 to the truth of the representations and authenticity 

 of the eggs, developed into positive disbelief; and 

 Bonaparte himself recanted, and took that decidedly 

 retrograde course, which others had entered upon, of 

 associating the monotremes with the marsupials in 

 the unnatural and artificial negative group of Ovovi- 

 vipara, or lmplacentalia. I, too, was so far influ- 

 enced by the prevalent scepticism or disbelief, and by 

 the similiarity of the monotieme egg to that of a rep- 

 tile, that I retained viviparity as a special attribute of 

 the mammals in 1872, although 1 declined, on other 

 evidence, to include a small size for the eggs in my 

 diagnosis of the class. I then, also, adopting the sub- 

 classes Monodelphia, Didelphia, and Ornithoclelphia, 

 segregated them into the major groups, combining 

 the first two under the name Eutheria, and contrast- 

 ing the last as the Prototheria. These names have 

 since been accepted by Professors Huxley, Flower, and 

 others; and, inasmuch as Professor Huxley did not 

 accredit their origin, they have been ascribed to him. 

 I must add, however, that Professor Huxley has 

 restricted the name Eutheria, although apparently 

 with a hypothetical qualification, to the monodelphs, 

 while he has coined a new name (Metatheria) for the 

 marsupials. I fail to appreciate the need for such 

 modifications, which virtually become exact syno- 

 nymes of Monodelphia or Placentalia, and Didelphia. 

 Finally, the old data as to the oviparity of mono- 

 tremes became almost lost to memory, so that no one 

 has recalled them since the rediscovery. In view of 

 such forgetfulness and scepticism, therefore, further 

 information was necessary to insure the admission of 

 the old evidence as valid. But Mr. Caldwell has 

 further added the intelligence, quite new, that the 

 eggs of Ornithorhynchus are meroblastic. This dis- 

 covery will have an important bearing on the question 

 of the origin of the mammals, and is antagonistic to 

 the suggestion of Professor Huxley that the type was 

 a direct derivative from the amphibians, while it in- 

 creases the possibility that Professor Cope may be 

 nearer the truth in affiliating the ancestors of the 

 mammals to the theriomorphous reptiles of the Per- 

 mian. Theo. Gill. 

 Sun-spots. 

 The long-delayed maximum of solar spots, now 

 undoubtedly passed, has attracted unusual attention 

 to the spot-periodicity. To-day and yesterday the 

 visible hemisphere of the sun was, for the first time 

 in nearly fourteen months, observed to be entirely 

 free from spots ; the occasion next preceding this 

 being 1883, Sept. 25. During the past two years, the 

 only additional days on which the sun was observed 

 to be without spots were, in 1882, Oct. 9 and Dec. 3, 

 and, in 1883, Feb. 23, and May 25, 26, 27, and 28. 



David P. Todd. 



Lawrence observatory, Amherst, Mass., Nov. 8. 



The numerical measure of the success of 



predictions. 

 Suppose we have a method by which questions of a 

 certain kind, presenting two alternatives, can in every 

 case be answered, though not always rightly. Sup- 

 pose, further, that a large number of such answers 

 have been tabulated in comparison with the events, 

 so that we have given the following four numbers : — 

 {aa), the number of questions for which the answers 



were the first way and the events the first way; 

 {ab), the number of questions for which the answers 



were the first way and the events the second 



way; 

 (ba), the number of questions for which the answers 



were the second way and the events the first 



way; 

 (bb), the number of questions for which the answers 



were the second way and the events the second 



way. 



Then the problem is, from these data to assign a 

 numerical measure to the success or science of the 

 method by which the answers have been produced. 

 Mr. G. K. Gilbert (Amer. meteorological journal, Sep- 

 tember, 1884) has recently proposed a formula for this 

 purpose ; and I desire to offer another. 



I make use of two principles. The first is, that any 

 two methods are to be regarded as equal approxima- 

 tions to complete knowledge, which, in the long-run, 

 would give the same values for (aa), (ab), (ba), and 

 (bb). The second principle is, that if the answers 

 had been obtained by selecting a determinate propor- 

 tion of the questions by chance, to be answered by an 

 infallible witness, while the rest were answered by 

 an utterly ignorant person at random (using yes and 

 no with determinate relative frequencies), then the 

 approximation to knowledge in the answers so ob- 

 tained would be measured by the fraction expressing 

 the proportion of questions put to the infallible wit- 

 ness. The second witness may know how often he 

 ought to answer 'yes;' but I give him no credit for 

 that, because he is ignorant when he ought to answer 

 'yes.' 



Let i be the proportion of questions put to the in- 

 fallible witness, and let j be the proportion of ques- 

 tions which the ignorant witness answers in the first 

 way. Then we have the following simple equa- 

 tions: — 



(aa) = i j (aa) + (ba) j + (1 — i)j j (aa) + (ba) j , 



(ab) = (l-i)j\(ab) + (bb)\, 



(ba) = (1 -i) (1 -j)\(aa) + (ba) \ , 



{bb) = i j (ab) + (bb) } + (1 - i) (1 -j) j (ab) + (bb) i . 



Now, whatever the method of predicting, these equa- 

 tions can always he satisfied by possible values of i 

 and j, unless the answers are worse than if they had 

 been taken at random. Consequently, in virtue of 

 the two principles just enunciated, the value of i 

 obtained by solving these equations is the measure 

 of the science of the method. This value is, 



(aa) 



(aa) + (ba) 



(aa) 



(ab) 



4- 



(ab) + (66) 

 (66) 



- 1, 



(aa) + (ba) ' (ab) + (66) 

 (aa) (bb) — (ab) (ba) 

 " \{aa) + (6a) j \(ab) + (66) j ' 

 Mr. Gilbert's formula has the same numerator, but 



