274 



NATURE 



\Juiy i.9, 1888 



The probability of an event is the value of the expecta- 

 tion of its occurrence existing in the mind of the thinker : 

 " We must again warn the reader that probabilities are 

 in his mind, not in the urn from which he draws" (De 

 Morgan, " Enc. Met.," 414) ; but in the solution of these 

 problems this subjective value is converted with startling 

 ease into a much more objective and concrete expression. 

 As Forbes puts it, " The doubt existing whether an event 

 still future, which may happen in many different ways, 

 shall occur in one particular way is not equivalent to 

 an inherent improbability of its happening, or having 

 happened, in that way " 



We do not assume that a friend is speaking untruly 

 when he tells us that, out of 10001 seats, the number of 

 his ticket is 453, yet the antecedent probability is 1/10000 

 against the truth of his statement. The chances are 

 greatly against ten stars out of 230 appearing as binary 

 combinations ; but, according to one view of the meaning 

 of " random distribution/' that arrangement is no more 

 unlikely than any other, and we should be no more 

 surprised to hear that one rather than another is the 

 actual one. Forbes objects that " to assume that ' every 

 star is as likely to be in one position as another,' is not 

 the expression of the idea of random or lawless distribu- 

 tion." . The expression seems to me to be true, but its 

 interpretation into mathematical symbols has been far 

 too closely restricted both by Michell and Forbes. 



" Michell assumes that, with random distribution, the 

 chance of finding a star in a space is proportional to the 

 space, or that a perfectly uniform distribution would be 

 that alone which would afford no evidence of causation." 



Suppose the whole surface of the sphere cut up into 

 minute equilateral triangles, and a star placed at each 

 collection of angular points. Each star is the middle 

 point of a regular hexagon, and at a distance, a, from six 

 other stars. If we imagine the six stars to be fixed, and 

 the central star shot out from the centre of the sphere 

 so as to fall within the hexagon, that it may not fall 

 within a distance, r, of any other star it must fall in a 

 regular hexagon, the side of which is (a — r) situated 

 symmetrically within the larger hexagon. The prob- 

 ability of the star falling within this smaller hexagon is 



expressed bv 5 '- , which becomes less and less the 



a 1 



more nearly r equals a ; that is, the more nearly the dis- 

 tribution is truly uniform. When r = a, the expression 

 becomes o, or the probability of exactly uniform distribu- 

 tion is nil, and apparently uniform distribution is due 

 solely to the imperfections of our instruments. Michell, 

 however, seems to assume this probability to be 1, or 

 certainty. Struve's method is open to the grave objec- 

 tion that he assumes that the total possible number of 

 binary combinations really occur. Applying his formula 

 to calculate a value for it which makes the chance a 

 certainty, we find that, if 2917 stars are scattered over 

 the sphere, it is a certainty that each will be vvithinf 

 3' 20" of another ! Of the three methods, that of Forbes 

 seems to be the least open to objection. 



Besides these fundamental difficulties in principle, 

 there are several very doubtful points in the calculation 

 which may be worthy of a brief notice. 



Michell considered the whole surface of the sphere, 

 though in his time the examination of the southern hemi- 

 sphere was hardly complete enough to furnish the requisite 

 data. The stars do not lie on the surface of a sphere, but 

 scattered through infinite space, so that two stars, the 

 angular distance between which is apparently small, 

 may in reality be very far apart. Suppose that the 

 nearer star lies on the surface of our imaginary sphere, 

 the probability that the direction of the other star is 

 within 1 5 of the surface is only about one-fourth. Hence 

 the number of apparently double stars must be reduced 

 to a considerable but unknown extent. 



Forbes throws considerable doubt on the correctness 

 of raising a second time to the power n. Struve's multi- 

 plication by «'s seems to prove very curious conclusions. 

 Mr. Venn's reasons for dissenting from Michell's solution 

 will be found well worthy of perusal (" Logic of Chance," 

 p. 260). Sydney Lupton. 



VEGETABLE RENNET. 



'"P HE idea that the protoplasm or living substance of 

 -*• both animals and plants is essentially similar, if not 

 quite identical, has long been accepted by both physio- 

 logists and botanists. This similarity is most easily seen 

 in the very lowest members of both kingdoms ; in fact, 

 for a very long time doubt existed in the case of many 

 organisms — eg. Volvox — as to which kingdom they 

 should properly be included in. Even now it is hardly 

 possible to formulate a definition of " plant " or " animal ;: 

 which shall put all into their proper positions. When we 

 go higher up the scale in both the animal and the 

 vegetable world, this difficulty of course disappears, on 

 account of the differences of organization and develop- 

 ment. It is not difficult even here to trace a remarkable 

 similarity of properties in the living substance, which 

 leads to the conception that not only is protoplasm 

 practically the same in animal and vegetable, but that its 

 activities in the two cases — that is, the metabolic pro- 

 cesses which accompany, and are in a way the expression 

 of, its life — are fundamentally the same. In both king- 

 doms we have as the sign of its life the continual building 

 up of the living substance at the expense of the materials 

 brought to it. as food, and the constant breaking down of 

 its substance with the consequent appearance ot different 

 organic bodies, which are strictly comparable in the two 

 cases. The vegetable protoplasm produces starch, the 

 animal glycogen — both carbohydrate bodies of similar 

 composition and behaviour. In both organisms we meet 

 with sugars of precisely similar character. The proteid 

 bodies long known to exist in animals, and classed into 

 albumins, globulins, albumoses, peptones, &c, have been 

 found to be represented in vegetables by members of the 

 same groups, differing but in minor points from them- 

 selves. We have fats of complex nature in the animal 

 represented by oils of equal complexity in the vegetable, 

 their fundamental composition being identical ; even the 

 curious body lecithin, so long known as a constituent of 

 nervous tissue in the animal, having been procured from 

 the simple yeast plant. 



Further, the changes which give rise to these bodies, or 

 which bring about various transformations of them, have 

 been in very many cases demonstrated to be due to 

 similar agencies at work in both the animal and vegetable 

 organism In many cases, no doubt, they are produced 

 by the actual splitting up of the protoplasm itself ; but 

 apart from this we have their formation in large quanti- 

 ties by the agency of bodies which are known as unor- 

 ganized ferments, and which are secreted by the proto- 

 plasm for the purpose of such formation. Perhaps no 

 line of research in vegetable physiology in recent years 

 has been so productive of good results as the investiga- 

 tions that have been made into the occurrence of such 

 bodies, and the comparison of them with those that are 

 met with in the animal organism. Diastase in vegetables, 

 and the ferments of saliva and of pancreatic juice in 

 animals, possess the same power of converting starch into 

 sugar. The peptic and tryptic ferments of the stomach 

 and pancreas respectively have been shown to have 

 representatives in the vegetable kingdom, and these not 

 only in such cases as the carnivorous plants, but to 

 actually made use of in such truly vegetable metabclisr 

 as the processes involved in the germination of the seec 

 The conversion of albumins and other indiffusible pre 

 teids into a further stage than that of diffusible peptone- 



