458 



NATURE 



lAIarc/i 15, 1888 



and simplifying calculations involving contingencies de- 

 pending on several lives. The hypothesis of Gompertz, 

 as formulated by Makeham, is, no doubt, useful for 

 graduating certain tables, and for dealing with some of 

 the more complex problems of hfe contingencies, but we 

 doubt whether a disproportionate consideration is not 

 given to it and to its application. In so far as it presents 

 itself to us as the most successful effort yet made to 

 fasten down the law of mortality, it has, no doubt, a 

 charm and a fascination for everyone, and especially the 

 mathematician ; but, keeping in view the limited use 

 made of it for the ordinary purposes of assurance work, and 

 that even for graduating it is only one of several methods 

 in vogue, we are inclined to think a less elaborate treat- 

 ment would have been more commensurate with the 

 proper scope of a text-book and book for general 

 reference. 



The next, and of course the main, portion of the volume 

 is concerned with the great class of questions involving 

 the consideration of interest when combined with life 

 contingencies ; that is to say, with annuities and assur- 

 ances, whether on single or joint lives, and whether abso- 

 lute or contingent ; with advowsons, next presentations, 

 fines for the renewal of leases on lives ; also with life 

 interests and reversions, and the values of life policies. 

 Explanations and demonstrations are given at length, 

 and some of them are exceptionally good. We may note 

 that, in the chapter on annuities and assurances, the 

 author says : " It has been common, in treatises on life 

 annuities, to deal with annuities and assurances separ- 

 ately, but the two classes of benefits are so intimately 

 connected that they ought always to be taken together." 

 We are not quite sure that we have caught exactly the 

 nature of the objection entertained by the author to the 

 common method of dealing with the two kinds of benefit. 

 We take it the intimate connection alluded to implies that 

 both things are built up of the elementary forms of which 

 V'lx + n is the type, and proceed on parallel hnes, and 

 not that the results for the one should be obtained by 

 giving an algebraic twist to the results deduced by a direct 

 process for the other. We do not infer from his words, 

 or gather from his book, that he would not exhibit the 

 present value of an assurance by direct reference to the 

 present value of ^l to be received by each of the persons 

 alive at age .i-+«, rather than obtain it by an indirect 

 process of reasoning, such, for instance, as this : — " If 

 here be an annuity on (.r) payable at the end of each year 

 on which he enters, and another annuity payable at the 

 end of each year which he completes, it is evident that 

 the difference between the two is the value of ;i^i payable 

 at the end of that year on which (.v) enters, but which he 

 does not complete ; that is, the value of ^i payable at 

 the end of the year of the death of {x), or, in other words, 

 the value of an assurance on (.f). Now v{\ -\- a.xj is evi- 

 dently the value of the first-named annuity, and, deduct- 

 ng from this the value of the ordinary annuity, a^, we 

 have the value of the assurance, t'(i + a^) — ax" The 

 building up of a formula by premising its verbal inter- 

 pretation is often an admirable example of ingenuity, but 

 this process can never be allowed to displace the estab- 

 lished course of mathematical reasoning. 



In this, the staple portion of his work, Mr. King manifests 

 bis extensive acquaintance with the subject, or, rather, 



subjects. With a great quantity of matter at his com- 

 mand, he has used the pruning-knife very sparingly, 

 possibly too sparingly. All the usual formulae are given 

 for precise calculation, and a number of approximative 

 processes are developed where an exact calculation would 

 be too cumbrous for actual use. It is worth suggesting 

 for consideration whether a collection of questions to be 

 worked out by students might not with advantage be in- 

 serted in a future edition of the book. There are many 

 precedents for such a course in connection with text- 

 books, and a goodly supply of questions is already at hand 

 in the examination-papers set at the Institute in past 

 years. 



There is a third portion of the work, occupying some 

 seventy pages, in which finite differences, interpolation, 

 and summation are treated with more fullness than 

 branches of pure mathematics would seem to be entitled 

 to in a volume professedly assigned to life contingencies. 

 Indeed, the author admits in his preface that these 

 subjects were not within the scope of the text-book as 

 originally planned. No doubt we have placed before us 

 propositions which are specially applicable to actuarial 

 needs, arranged and demonstrated with Mr. King's usual 

 ability ; but it seems to us they would have been more 

 conveniently published in some other connection than the 

 present. A knowledge of .these things in a duly regulated 

 course of study would naturally precede the consideration 

 of their application. 



The text of the work is supplemented by a collection of 

 interesting tables, commencing with a table of mortality 

 based on a combination of data for young and mature 

 lives, and intended to show the mortality of healthy male 

 life from birth to extreme old age. We must not fail to 

 mention that the collection embraces complete tables for 

 finding the value of joint-life annuities up to four lives 

 inclusive. 



Looking at the work as a whole, we find the various 

 subjects are cleverly handled, the propositions appear one 

 after the oLher in well-ordered succession, the demon- 

 strations are well chosen, and the wording is clear and 

 effective. Altogether Mr. King has done his work dili- 

 gently and with good judgment, and has placed all future 

 students of the Institute under a debt of obligation to 

 himself and to the Council. 



ROSENBUSCH'S '' PETROGRAPHY^' 

 II. 

 Mikroskopische Physiographie der massigen Gesteine. 

 Von H. Rosenbusch. II. Abtheilung. Zweite ganzlich 

 umgearbeitete Auflage. (Stuttgart, 1887.) 



IN a notice (Nature, vol. xxxv. p. 482) of the first part 

 of the present work, we showed that the author, adopt- 

 ing a natural system of classification which gives the first 

 place to field-evidence, divides the eruptive rocks into three 

 great groups, viz. (i) the Plutonic rocks ; (2) the Dyke 

 rocks {Ganggesteine) ; and (3) the Volcanic or Effusive 

 rocks. Unable to free himself entirely from the idea that 

 geological age ought to be an essential factor in rock- 

 classification, he subdivides the third group into 2kpal(eo- 

 volcanic and a neo-volcanic series. It is the treatment of 

 the neo-volcanic series which constitutes the bulk of this, 

 the second and final part of the book. 



