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 LIBRARY 
 
 OF THE 
 
 University of California. 
 
 GIFT OF 
 
 Class 
 
 
 
 
 
THE DEVELOPMENT 
 
 OF 
 
 INSURANCE MATHEMATICS. 
 
 TWO LECTURES 
 
 Delivered before the Students in the School of Commerce 
 of the University of Wisconsin, the Fall Term of 1901. 
 
 BT 
 
 MILES MENANDER DAWSON, Consulting Actuary. 
 
 FIRST LECTURE, 
 Preliminary Development, Decimal System, Series, Logarithms, Per- 
 mutations and CJombinations, Probabilities. 
 
 SECOND LECTURE. 
 
 Evolution of Actuarial Science, Annuities Certain, Mortality Tables, 
 Actuarial Science Proper. 
 
 1901. 
 THE INSURANCE PRESS, 
 
 Nuw YOBK. 
 

THE DEVELOPMENT OF INSDRAHCE MATHEMATICS. 
 
 FIRST LECTURE— PREIilMLN-ARY DEVELOPMENT. 
 
 It is most interesting to study the growth and evolution of any 
 branch of science and to observe how it sprang from some other 
 branch of science, and what things had to be known as a prepara- 
 tion for the dawning of the new learning. It is a far cry, perhaps, 
 from actuarial science to the decimal system of notation, and yet, 
 had it not been for the invention of that system it may well be 
 doubted whether arithmetic would ever have reached the stage 
 which renders mathematical problems that would otherwise be 
 difficult of operation, simple and easy. Addition, which is now 
 so simple, albeit so laborious a matter, was exceedingly difficult 
 under the old systems of notation. A student may readily form 
 some conception of the difficulties if he will range a number of 
 Eoman numerals side by side, or one under the other — one ar- 
 rangement is about as convenient as the other — and proceed to 
 add. Multiplication, which is continued addition, was even harder. 
 It follows that operations which must be performed upon a very 
 large scale, in order that actuaries may do their tasks, were then 
 almost impossible upon anything but a limited scale. 
 
 Almost equally absurd it may at first seem to declare that the 
 new science could not have come into existence until after algebra 
 was discovered; but it will be seen to be perfectly reasonable to 
 say this when it is taken into account that actuarial science is 
 merely a department of algebra, and that all its operations are 
 algebraic. The form which all its formulas take is that of equa- 
 tions, and they are evolved from other equations by purely alge- 
 braic processes. 
 
 Algebra and decimal notation were both introduced into Europe 
 by the Arabs through the Moorish schools of Spain and Africa, m 
 the thirteenth century. The first mathematical books to be stud- 
 ied were those of the Arabian authors and Arabian translations of 
 
 iionqi 
 
4 DEVELOPMENT OF 
 
 Euclid, Archimedes, Appolonius and Ptolemy. The stndy was for 
 a long time condemned by the church authorities, although one 
 of the popes was a mathematician. It was connected in the minds 
 of the people with astrology and alchemy and, indeed, most of 
 the early mathematicians were astrologers also. 
 
 The following things needed to develop in mathematics before 
 actuarial science could begin its evolution: Decimal notation, 
 decimal fractions — desirable, not necessary — series and summa- 
 tion, logarithms, permutations and combinations, probabilities. 
 
 Decimal fractions were brought in by Pitiscus in 1617. Their 
 use, however, did not become common until after 1700 and, indeed, 
 so little was known for a long time of the discovery of Pitiscus that 
 the honor of the invention was claimed by mathematicians of a 
 later date. 
 
 Series were treated by the ancient mathematicians, but in a 
 very fragmentary and imperfect manner, algebra being unknown. 
 Euclid and Appolonius, among others, discussed the subject some- 
 what, but it was never given the attention that was necessary in 
 order to make the development of annuities possible until in the 
 seventeenth and eighteenth centuries, when Leibnitz, Montmort, 
 James (or Jacob) and John Bernoulli, La Place, Gauss and Cauchy 
 threshed out the matter thoroughly. 
 
 Even with a perfected decimal system, both of integral and 
 fractional notation, and in spite of the perfection of algebra as a 
 mathematical art, it would have been most difficult to perform the 
 operations required in compound discount computations, as in an- 
 nuities certain, without the aid of logarithms, which make it al- 
 most as easy to find the present value of a sum due in 1,000 years 
 as if the term were a single year. In no branch of mathematical 
 science, perhaps, has the invention of logarithms proven a greater 
 blessing. Their invention, as is well known, is due to Napier, and 
 dates from about 1610. They were soon afterward improved by 
 Briggs, who also suggested the decimal base, which made logar- 
 ithms much easier to employ and far more useful. 
 
 Annuities certain grew so inevitably out of the formulas for the 
 summation of geometric series that it is impossible, perhaps, to 
 mark out closely the period when the first mention of the subject 
 is found. Indeed, although the topic has a deal of interest of itself. 
 
INSURANCE MATHEMATICS 5 
 
 and is generally considered abstruse and difficult by students, 
 problems in annuities certain, like all compound-interest prob- 
 lems, were originally merely set as exercises in series, and no 
 thought was given that such might be the foundation for a new 
 branch of the science. As a separate branch of mathematical re- 
 search, we do not find that annuities certain appear before the 
 beginning of the eighteenth century. 
 
 As series and, perhaps, logarithms, were necessary first steps to 
 annuities certain, which is one of the three legs upon which actu- 
 arial science will be found to stand, so in like manner permuta- 
 tions and combinations had to precede probabilities, which is the 
 second of the three supports of actuarial science. 
 
 Perhaps, as the subject first appeared in mathematics, nothing 
 was ever more unpromising so far as practical results were con- 
 cerned than permutations and combinations. That it could ad- 
 vantage anybody to know how many different arrangements could 
 be made of n things taken k at a time, seemed too ridiculous to 
 talk about. Often has it proved, however, that the topic which 
 was considered mere mathematical sport has been found most 
 useful in some field which was awaiting that very discovery to un- 
 lock new treasures of practically valuable learning. So it was in 
 the case of the apparently idle speculations as to the numl er of 
 arrangements of different things which one could make; what 
 seemed to the schoolmen only elegant and frivolous speculation, 
 highly amusing because of the frequently startling results, was tho 
 missing word to unseal the secrets of probabilities. 
 
 The subject was not unknown to the ancients, either of Greece 
 or of India. Aristotle, whose death took place 322 B. C, suggestea 
 the solutions of such problems, and, before his time, Arjabhatta,, 
 the Indian savant, treated certain phases. The modern develop- 
 ment began in the later years of the sixteenth century, in the re- 
 searches of Cardan, which, however, were not printed until 1663. 
 He was followed soon by Pascal, Leibnitz and James (or Jacob) 
 Bernoulli. 
 
 Concurrently with the development of the theories of permu- 
 tations and combinations came their application in the science of 
 probabilities. For Cardan, who died in 1576, wrote the first known 
 essay on the subject of probabilities, though the same was not pub- 
 
6 DEVELOPMENT OF 
 
 lished until 1663. It was known by the title "De ludo aleae" — 
 ^^Concerning the game of dice." 
 
 In ancient times they evidently had the conception that the 
 value of a chance or probability could be valued; for, as we shall 
 see, the courts occasionally set values upon life annuities. More- 
 over, loans were made by venturesome capitalists upon ships and 
 cargoes, repayable at more than the usual interest, but forgiven 
 entirely if the property were destroyed. The similarity to insur- 
 ance is clear, and it is also evident that there was some rough 
 fashion by which what was believed to be the value of the hazard 
 was ascertained. But if, as frequently in our days, there was 
 empirical guessing at the value of hazards, there was no attempt 
 made to calculate these values, so far as is now known. 
 
 Galileo, at some time during his life, which closed in 1642, also 
 wrote briefly upon the same subject as Cardan. 
 
 But the first work of importance consists of a series of letters 
 by Pascal in reply to certain inquiries of the Chevalier de Mere — 
 by some thought to have been mythical — concerning the chances 
 of play. The first of these letters was written in 1654, and they 
 were published in 1679. In them some of the most intricate 
 problems of probabilities have been worked out. The philosopher 
 was not satisfied to deal merely with the simple questions that 
 lay u- -on the surface, but boldly assailed the most complicated. In 
 the course of his investigations he discovered the "arithmetical tri- 
 angle'^ and "figurative numbers," which have puzzled the drones 
 ;.mong college students from that day to this, but have served 
 nany good purposes in mathematics, and stood ready to welcome 
 ihe binomial theorem at a later date. Merely to mention this one 
 discovery, sot forth in his letters, may serve to indicate that he 
 did not indulge in mere child's play in his primer of probabilities. 
 His contemporary, Fermat, also took part in the discussion. 
 
 The next work upon the subject was by Christian Hu5^gens. 
 Like the first writing dealing with probabilities, it was an attempt 
 to compute the chances with dice. Its title was "De Eatiociniis 
 in Ludo Aleae." The pamphlet appeared in 1657. 
 
 After this there was a long pause. The situation in mathemat- 
 ics was at the time about as follows : After the subject had been 
 reintroduced it was for a long time strenuously opposed by church 
 
INSURANCE MATHEMATICS 7 
 
 influences. Moreover, it was by the schools despised as forming no 
 part of a liberal education. The universities for a long time knew 
 it not. The connection of mathematics with astrolog}' — all who 
 studied the stars were esteemed astrologists in those days — ^with 
 alchemists — all students of nature were considered alchemists — 
 did not add to its repute. The church was suspicious of all 
 learning which did not emanate from itself. But at about this 
 time, as a result of the serious attention to mathematics, which 
 some of the greatest minds of the age had paid, and also, perhaps, 
 because the control of education was passing into secular hands, 
 mathematics was beginning to make its way into the universities. 
 But this was only in the more abstract forms. Most of its appli- 
 cations to chemistry, astronomy and other sciences were yet to 
 be made. The least promising of these applications seemed to be 
 to probabilities ; for not only was it denied that there was or could 
 be such a science, but the use of it, unless to encourage gaming, 
 nobody could see. Therefore the first attempts to solve problems 
 of chance were ridiculed by most scholars, who denied both that 
 it could be done and that it would be of any use if it were done. 
 Doubtless, therefore, the science would have been a long while 
 getting a start, had not a recluse like Pascal, fond of all specula- 
 tions that tested his skill in analysis and synthesis, and altogether 
 careless whether his speculations were useful or not, been tempted 
 to essay the task of estimating chances. 
 
 Notwithstanding this auspicious beginning, however, there was 
 nothing further upon the subject for nearly forty years, and noth- 
 ing of much consequence for more than fifty years. In 1693 the 
 first study of the subject appeared in English; its title is "An 
 Arithmetical Paradox Concerning the Chances of Lotteries." Thus 
 but little over two hundred years ago actuarial science, which was 
 first brought to its full development in Great Britain, found lodg- 
 ment there. 
 
 This first English work was of minor importance. It merely 
 treated some practical phases of the subject, and was by no means 
 thorough. But there had been for many years preparing a really 
 great work upon the subject, the first of all great works. "Ars 
 Conjectandi," by Jacob (often called James) Bernoulli, was, of 
 course, completed before the author's death in 1705, but it was 
 
8 DEVELOPMENT OF 
 
 not brought out until 1713. It was edited and published by the 
 nephew of the author, Nicholas, son of John Bernoulli; and from 
 this date the subject of probabilities afforded this family opportu- 
 nities to distinguish themselves even unto the third generation. 
 
 That "Ars Conjectandi" marks the close of an old era and the 
 beginning of a new is sufficiently attested by the following: It 
 was written in Latin, a language until then practically always 
 employed as a vehicle for learning, and it dealt with the newest 
 application of what was itself a new science, algebra. It was at 
 the turning point. Latin was going out; science was coming in. 
 The book was not a mere sketch or study. Jacob Bernoulli had 
 toiled during the last years of his life most indefatigably and 
 without paying the least attention to the circumstance that the 
 whole speculation was by most learned men deemed vain and un- 
 profitable. When he laid down his pen, disappointed even in his 
 hope that his manuscript should be published, he had developed 
 the scraps of speculation into a demonstrated science. And al- 
 though even at a later day persons could be found who were so 
 dull as not to apprehend his truths, and so intensely prejudiced as 
 to decide against him on a priori and most fanciful grounds, he 
 had fixed the fundamental lines for the investigation of probabili- 
 ties where they have ever since been found. His work dealt with 
 all the simpler problems and with many of the compound problems, 
 including some of the most intricate. It was long the storehouse 
 from which later mathematicians drew their material, and what is 
 owing to Bernoulli it would now be very difficult to overestimate. 
 He found probabilities a subject for occasional and wholly tenta- 
 tive reasoning; he left it a science. 
 
 After the death of Jacob Bernoulli, but before the publication 
 of his book, there appeared in France, in 1708, a pamphlet entitled 
 "An Essay on the Analysis of the Game of Hazard." The author 
 was the great Montmort. The work has value, but need only be 
 here mentioned to show that the speculations of Pascal had not 
 been fruitless in his own country. There for some time the topic 
 was, however, following Pascal's lead, treated only as to the proba- 
 bilities in various games of chance. 
 
 The next great mathematician to take up the matter was also 
 a Frenchman, De Moivre; but he was an exile in London, and 
 
INSURANCE MATHEMATICS 9 
 
 his books were written in English. His fame was also achieved 
 in England, and he is known as a great English mathematician. 
 I have said that his books were written in English, which also was 
 the case as to all, save a pamphlet entitled "De Mensnra Sortis," 
 which was published in 1711. This was followed by his great 
 work, ^T)octrine of Chances," which was published in 1718, and 
 was the second book to treat the entire subject of probabilities, 
 and the first to appear in a modern language. This book re- 
 mained the standard text -book upon the subject for a long time, 
 and during the author's lifetime two revised editions, the last 
 much enlarged, also appeared. 
 
 Meanwhile the Bernoullis were keeping interest in the subject 
 alive in other countries. Nicholas Bernoulli even preceded the 
 publication of his uncle's "Ars Conjectandi" by his o^vn pamphlet, 
 '•'Specimens of the Art of Conjecture, Applied to Questions of 
 Law," which was printed in 1709. Daniel Bernoulli, his brother, 
 followed this in 1738 with his "Specimens of a New Theory Con- 
 cerning the Measurement of Chance." 
 
 In Great Britain, De Moivre reigned supreme in this branch 
 of applied mathematics for many years. He enjoyed great repute 
 and, as we shall see when we discuss actuarial science itself, he was 
 in great demand to put into practice his own theories, in the form 
 of computations of probabilities of various sorts. But in 1740 a 
 rival arose, one Thomas Simpson, a self-taught mathematician, 
 who had also been a necromancer — a trade by this time out of 
 fashion in connection with mathematics — and who at first enjoyed 
 no standing whatever, although before his life closed his accom- 
 plishments were such that the Encyclopedia Britannica gives him 
 credit as the greatest non-academic mathematician that Great 
 Britain has produced. Simpson's first book, published in 1740, was 
 entitled "The Nature and the Laws of Chance." It was written 
 in language as simple as the author could employ, because it was 
 his purpose to render the subject popular and easy of comprehen- 
 sion by the ordinary untrained mind. This of itself was a twofold 
 offense against De Moivre, because it was, in the view of the dis- 
 tinguished mathematician, both an attempt to dispute his unri- 
 valed position and also an attempt to cheapen and lower his sci- 
 ence. Consequently the younger man, who, as events have proved, 
 
10 DEVELOPMENT OF 
 
 deserves more credit for the growth and extension and perfection 
 of actuarial science and practice than even De Moivre, was by no 
 means welcomed by his great contemporary. 
 
 In France the discussion of probabilities yet turned upon par- 
 ticular hazards in games of chance. Thus, so late as 1751, we find 
 the great Euler, one of the most wonderful mathematicians of all 
 time, sending forth a pamphlet entitled "Calculations of the 
 Probabilities of the Game of Eencontre." The time when France 
 would again make a great contribution to the science of probabili- 
 ties was thus yet a while deferred. 
 
 In 1773 La Grange began a work on "An Article on Probabil- 
 ity," but it was many years before the subject was treated exhaust- 
 ively by a Frenchman, which seems all the more remarkable since 
 France had so many able mathematicians, and also since the first 
 work of importance upon the subject, Pascal's letters, appeared 
 in that tongue. But the slowness with which the science of proba- 
 bilities in general developed there may, as we have seen, be as- 
 cribed with much plausibility to the circumstance that Pascal's 
 letters were concerning certain peculiar gambling hazards. This 
 view is borne out by the fact that, while discussions of probabilities 
 in general were missing, discussions upon particular problems of 
 gambling were frequent. 
 
 Between 1750 and 1760 three Englishmen contributed short 
 but valuable works upon the subject, Hoyle in 1754, James Dodson 
 in 1755, and Samuel Clark in 1758. In 1769 John Bernoulli, 
 grandson of John Bernoulli, the elder, and grand nephew of Jacob 
 Bernoulli, published a pamphlet, the last to bear the distinguished 
 name which was attached to the first comprehensive work upon 
 the subject. 
 
 France, which had so long dragged behind Great Britain in 
 developing this science, began to bestir herself in the last years 
 of the eighteenth century when, by reason of the impulse which 
 the revolution gave all men, sciences and arts began to develop 
 as never before. The first fruit of this travail was a pamphlet 
 by Condorcet in 1785. But it reached its full development when, 
 in 1812, La Place's "Analytical Theory of Probabilities" appeared. 
 This book remained the classic text-book upon the subject for 
 many years, and really exhausted all that was known concerning it. 
 
INSURANCE MATHEMATICS 11 
 
 besides containing much that was original and of great value. It 
 remains to this day one of the three or four really valuable books 
 upon the science. 
 
 Well might it be such, also, for not only was La Place one of 
 the greatest mathematicians of his own or of any time, but he 
 labored under an encouragement which had never before been 
 given any student of the subject. Pascal's letters brought forth 
 ridicule and abuse, almost without a dissenting voice, excepting 
 from advanced mathematicians who were as much out of favor as 
 himself. Much the bold speculator cared about that, however; 
 but his successors did care. For, be it known, the heterodoxy of 
 to-day becomes the orthodoxy of to-morrow as the world ad- 
 vances. When Pascal's work took its place in the accepted views 
 of men of learning the orthodox set their line just beyond what 
 Pascal had demonstrated. All beyond that was heterodox, and 
 therefore, as ever, the ban was put upon the innovator. In France, 
 too, there was rather more of this than elsewhere, though Jacob 
 Bernoulli's great work could not see the light until its author was 
 dead. In England alone, thanks to the mathematical turn which 
 Newton's great achievements gave academical education, the re- 
 ception of the new learning was warmer, but even there the one- 
 time innovator, De Moivre, threw all sorts of obstacles, not spar- 
 ing ridicule, in the way of Simpson. 
 
 With La Place it was different. Not only had the revolution 
 broken up the theological and scholastic prejudices in France and 
 installed instead a passion for innovation, but La Place received 
 the direct support of the emperor, was rewarded directly for his 
 services, and saw his book published with honors and at the na- 
 tion's expense. In turn he dedicated it to the emperor who thus 
 befriended him and encouraged science. Besides, La Place also 
 had access, of course, to all that had already been written upon 
 the subject, which had now been under the consideration of some 
 of the brightest minds for more than a century and a half, and 
 had been an established and demonstrated science for almost 
 exactly a century; for Bernoulli's "Ars Conjectandi" appeared in 
 1713, and La Place's "Analjiical Theory of Probabilities" in 1812. 
 
 Although after the publication of La PlacS's monumental work 
 the evolution of the science of probabilities may be considered to 
 
12 DEVELOPMENT OF 
 
 have been completed in all essential particulars, there remained 
 much to do in the matter of rendering the same lucid and readily 
 intelligible, and also of adapting it to the practical purposes for 
 which it was intended. This was the task of the nineteenth cent- 
 ury especially, as will more fully appear when we take up actuarial 
 science as a separate study, although this development also began 
 early in the eighteenth century. Most of these productions have 
 taken the form of works upon special applications of the science of 
 probabilities, but there have been at least three books upon proba- 
 bilities which deserve particular mention. 
 
 In 1838 Professor Augustus De Morgan published his "Essay 
 on Probabilities," which, notwithstanding the modesty of the title, 
 was a very thorough and complete work upon the subject, making 
 a thick volume. It became the standard text-book upon probabili- 
 ties in the English language, and held that place until this branch 
 of mathematics began to be taught in the universities as one of the 
 higher branches, when a work that is more technical, less diffuse, 
 with more formulas and less of explanation and reasoning, and, 
 above all, with many examples for solution, was demanded, and 
 was supplied in Whitworth's "Choice and Chance," published with- 
 in the present generation, and now the leading English text-book 
 upon the subject. In America, although in our country the princi- 
 ples of the science of probabilities have been more widely applied 
 in practical affairs than in any other country, nothing has been 
 done toward making text-books upon the subject, and very little 
 instruction upon it has been given in schools or colleges. 
 
 A work of the greatest importance upon the principles of the 
 science of probabilities, and upon their application to many things, 
 was published in French in 1845. It was the work of a Belgian, 
 M. Quetelet, and was entitled "Letters on the Theory of Proba- 
 bilities as Applied to Moral and Political Sciences." The writing 
 of these letters began in 1837, without any thought of publishing 
 the same. Their value and significance may be estimated, however, 
 by the fact that almost immediately upon their appearance thej 
 were translated into English and other languages, and became the 
 most valuable text-book upon the subject that its students could 
 turn to. This was very largely because of the fullness and clear- 
 ness with which the application of the law of probabilities to point 
 
INSURANCE MATHEMATICS 13 
 
 the lessons of statistics was set forth, though the same lucidity- 
 marked the whole of the work. 
 
 The history of the development of the science has thus far 
 been traced from the earliest times to the present. It is, as we have 
 seen, essentially a purely modern science. This branch of mathe- 
 matics could hardly develop in advance of algebra, decimal nota- 
 tion, decimal fractions, series, annuities certain and the like, for 
 much of it, in theory or application, depends upon one or all of 
 these. Singular is it, also, that the first great work upon the sub- 
 ject and the last, in the sense of an original and suggestive work, 
 the contributions of Pascal and Quetelet, have both been in the 
 form of letters, written with no intention originally of publishing 
 the same. We shall see, when we come to consider the origin of 
 mortality tables and the evolution of actuarial science, that the 
 growth of the body of knowledge concerning probabilities has been 
 accompanied by the application of the principles of the science to 
 the practice of insurance, and especially of life insurance. 
 
 It remains, then, only to discuss what this science of probabili- 
 ties is which we have been tracing through its different stages of 
 historic development. The primary discovery was that when a 
 thing had happened to a certain number out of a group, to each 
 of which it was a priori equally likely to happen, the probability 
 that it had happened to a particular one in the group, when one 
 did not know which, could be expressed by a fraction, of which 
 the number to whom it had happened was the numerator, and the 
 number in the whole group, that is, the number to whom it had 
 happened and to whom it failed to happen, combined, was the 
 denominator. This was the fundamental discovery. 
 
 Following this at no great distance was the discovery that the 
 chance that the thmg had failed to happen to a particular one in 
 the group could be measured by a fraction, with the number to 
 whom it had failed to happen as a numerator, and with the same 
 denominator as before. A little consideration sufl&ced to show that 
 these were complementary values, the sum of which was unity, 
 because adding them gave the number to whom it happened plus 
 the number to whom it did not happen, both for numerator and for 
 denominator. Thus it was determined that the sum of the chances 
 that a thing happens and that it did not happen is always unity; 
 
14 DEVELOPMENT OF 
 
 but, since such a summation also brings certainty, for as to a given 
 individual it may with confidence be affirmed that the thing either 
 has happened or has not happened, it has been customary to rep- 
 resent the mathematical value of certainty by unity. A very in- 
 genious proof that this is the correct thing, and indeed the only 
 thing to do, is given by Whitworth in "Choice and Chance," already 
 referred to; it will be seen that actual proof offers difficulties, 
 though it is not hard to see that the thing is true. The following 
 is the passage : 
 
 "But it may be asked, is certainty a degree of probability at all, 
 or can smaller degrees of probability be said to have any ratio of 
 certainty? Yes. For if we refer to the instance already cited of 
 the six passengers in the ship (from which one person had been 
 lost) we observe that the chance of the lost man being a passenger 
 is six times as great as the chance of his being our friend (one 
 of the six). This is the case, however great our ignorance of the 
 circumstances of the event ; and it will evidently remain true until 
 we attain some knowledge which affects our friend differently from 
 his fellow passengers. But the news that the lost man was a pas- 
 senger does not affect one passenger more than another. There- 
 fore, after receiving this news, it will still hold good that the 
 chance of the lost man being a passenger is six times as great as 
 the chance of its being our friend. But it is now certain that the 
 lost man was a passenger; therefore the probability that it was our 
 friend is one-sixth of certainty.'^ 
 
 Since by the fundamental rule of probabilities we will also find 
 that it is one-sixth of unity, it follows that certainty is correctly 
 represented by unity. 
 
 From these simple laws the science of probability extended to 
 compound probabilities. One of the most remarkable of the early 
 discoveries was that, when you know the chance that one thing 
 will happen, and also the chance that another will happen, the 
 chance that both will happen, unless one is prohibitive of the 
 other, is not the sum of the chances, but their product. And from 
 this the investigations proceeded to deal with all imaginable com- 
 pound and complex probabilities, many of the most difficult being 
 treated in advance of the development of principles for determin- 
 ing the simpler and easier values. 
 
INSURANCE MATHEMATICS 15 
 
 It was a long step, but one soon taken, from the probability 
 that a thing, known to have happened to one of a group, has 
 happened to a particular one, to the probability that events will 
 occur in future, a probability based upon averages, depending 
 upon their reliability, their extent, the correctness of classifi- 
 cation and their applicability. These problems have been so diffi- 
 cult in operation that to this day an enormous amount of the 
 theory of probabilities remains unappropriated by applied sciences 
 because of the great practical difficulties in the way of so apply- 
 ing them. Many very interesting subjects of speculation in the 
 laws of probabilities are therefore neglected by the actuary in 
 his practice, and, indeed, only the simpler elements of the science 
 are at the present time employed by him, though he would be a 
 rash man indeed who would undertake to say what will be the 
 limit of their practical application next year. 
 
 Though coming so late in the family of mathematical sciences, 
 probabilities already rejoice in a biographer of no mean reputation. 
 Todhunter, the great English mathematician, has written and 
 published an extensive work, now unhappily out of print, upon 
 the history of probabilities, in which he traces the growth of the 
 true theories, the correction of errors and the development of 
 new principles with great fullness and precision. His account 
 begins with the earliest writers upon the subject and closes with 
 La Place. From it much of the material for this lecture has been 
 obtained, and particularly so concerning some of the smaller and 
 less important pamphlets which are not to be found in the usual 
 library of the actuary. As to the more important, including even 
 Bernoulli's "Ars Conjectandi" and De Moivre's 'TDoctrine of 
 Chances," as well as La Place's great work, references to such 
 have been made directly. 
 
16 DEVELOPMENT OF 
 
 SECOND LECTURE— MORTALITY TABLES, ANNUITIES 
 CERTAIN AND ACTUARIAL SCIENCE PROPER. 
 
 We have seen how the fundamental mathematics, requisite 
 before actuarial science could develop, evolved; we have now to 
 consider the genesis and evolution of the application of the laws 
 of probability to insurance, and especially to life insurance. And 
 we shall find that, first of all, there needed to be something for 
 the law of probability to act upon, viz., a mortality table, and also 
 a handmaiden, in the form of a developed law of compound in- 
 terest and discount and of annuities certain. 
 
 First, then, as to mortality tables. We find the first of them 
 in times just back of mediaeval, and just this side of ancient. The 
 pretorian prefect, Ulpian, in 364 A. D., put forth a table of life 
 expectancies or probable lives, invented for the purpose of esti- 
 mating the value or annuities for probate purposes. The Eoman 
 law at that time provided that not more than a certain portion 
 of the value of an estate could be willed away from the heirs. 
 But testators left annuities, as charges upon their estates, the 
 same being 'payable to beneficiaries for life ; and so it became 
 necessary to estimate the value of these annuities. The Ulpian 
 method was simple. It assumed that a term of an annuity cer- 
 tain could be employed to express the value of a life annuity, and 
 that this term was that of the probable life of the annuitant, or 
 that term of years at the end of which the chances would be 
 equal that he was living or dead. 
 
 Actuarial science could have no real genesis until this fallacy 
 was dead and the truth had appeared. Ulpian's experiment was 
 forgotten, and his theory was wholly lost when we next encounter 
 something like a mortality table, and this time with true princi- 
 ples applied to ascertain the value of an annuity. The author was 
 no less a person that John de Witt, Grand Pensionary of Holland 
 and West Friesland, who turned aside from the onerous duties 
 of state to consider the subject of life pensions or annuities. Such 
 were already granted by states which were hard pressed for money 
 and which could not negotiate loans in the usual way. De Witt 
 formed the idea that the prices at which such were sold were ut- 
 
INSURANCE MATHEMATICS 17 
 
 terly inadequate, and he proceeded to investigate the subject. 
 His work was done before 1671, when the results of his labors 
 were presented in a state paper. This was not many years after 
 the appearance of the first pamphlets about probabilities, as, for 
 instance, Pascal's letters. It was more than forty years before 
 the publication, in 1712, of Jacob Bernoulli's "Ars Conjectandi," 
 the first comprehensive work on probabilities. Therefore, it may 
 be truly said, not merely that actuarial science began with de 
 Witt, but also that he built much of the substructure for it ; since 
 in his work he developed newly, if not first of all, the fundamental 
 principles of probabilities. 
 
 Most important of all else, though, in his work, was the dis- 
 covery of the true theory of the operation necessary to compute 
 the values of annuities; and this was: Take a large number of 
 annuitants at a given age, the mortality for each age being known, 
 and follow them down through life as they drop out and disap- 
 pear, computing the amount which would need to be paid each 
 year; reduce these sums to their present value; add them all 
 together, and you have the aggregate present value of all sums 
 payable on account of these annuities. The equal share of each 
 annuitant in this aggregate is manifestly the value of his annuity. 
 
 The value thus found is not the same that would be found by 
 treating the annuity as equivalent to an annuity certain for the 
 term of the expectancy, or of probable life, according to the same 
 table of mortality. But this value will prove by actually running 
 out, on the basis of the interest and mortality assumptions, while 
 the value from the annuity certain for a term equivalent to the 
 expectation or the probable life will not. Moreover, when we 
 come to examine into the matter, we cannot fail to be impressed 
 with the circumstance that the expectation could only furnish an 
 equivalent when there is no interest counted upon, for it is the 
 average of years of life ; while in effect, the value of the annuity is 
 the average of the same years of life, discounted. But the dis- 
 count is greater for payments which are longer deferred; from 
 which fact it appears that the equivalence no longer holds, when 
 discount is introduced. 
 
 The Ulpian theory died hard. It reappeared repeatedly after 
 actuaries had learned to deal with the subject according to science 
 
18 DEVELOPMENT OF 
 
 and reason. Thus in 1738, after the true principles were clearly- 
 enunciated and had been adopted by mathematicians generally, 
 a learned London barrister, Weyman Lee, published an ingenious 
 book in which he undertook to prove the equivalence. The name 
 of his book was "An Essay to Ascertain the Value of Leases and 
 Annuities for Years and Lives." The following passage is illus- 
 trative of his reasoning: 
 
 "I suppose no one will controvert these points : That he who 
 has an annuit}^ for the life of a person, has an annuity for such a 
 term of years as such person in fact shall live; and, when he buys 
 it, the term of years to which any person's life shall be prolonged 
 being uncertain, that he buys it for such a time as there is a 
 chance or reasonable probability that the person may live whose 
 life is nominated." 
 
 Lee also objected that the other theory made the value of 
 an annuity by the same mortality table equivalent to the values 
 for different terms of years, according as the interest assumed 
 was one rate or another. We have seen that this is necessarily 
 so, and that only when no interest is assumed is the value equal 
 that of an annuity certain for the expectancy. 
 
 The false theory, dating back to Ulpian, and so ably cham- 
 pioned by Lee, is a very natural generalization when one makes 
 a hasty conclusion, and it is certainly extremely plausible. As if 
 to emphasize that the human mind in its evolution ever travels 
 over the same track, we find this idea reappearing every now and 
 then by a strange sort of atavism, as it were, and in the most un- 
 expected quarters. Thus within the year a leading financial paper 
 of Xew York gravely announced editorially: 
 
 "Life insurance companies employ two factors in fixing pre- 
 miums on policies of insurance. They are, first, the average ex- 
 pectation of life of the insured person, according to carefully pre- 
 pared mortality tables; second, the rate of interest at which 
 money can be invested. The company takes the case of a man at, 
 say, 21 years of age, and finds from the mortality tables that his 
 average expectation of life is, say, 30 years. By simple arith- 
 metical calculations it computes the amount of money that such 
 a man should pay, annually, semi-annually, or quarterly, as the 
 case may be, in order that his money at the assumed rate of in- 
 
INSURANCE MATHEMATICS 19 
 
 terest, say 3J per cent, shall at least equal the amount of his policy 
 at the end of thirty years. The amount, thus computed, with an 
 allowance for expenses and profit, is his premium." 
 
 This apparent digression is really no digression at all. We 
 are to follow the growth of actuarial science, and it is worth 
 while, therefore, to know what it is, and particularly what it is 
 not, and why, therefore, John de Witt must be considered its 
 founder, and not Ulpian. 
 
 de Witt, in addition to discovering the fundamentally true 
 theory of annuities for life, and applying the principles of prob- 
 abilities and compound discount to the ascertainment of their 
 values, also invented a mortality table. This he constructed on 
 a plan of equal decrements, limited to from ages three to fifty, 
 however, followed by other periods in which the law of equal 
 decrement was also supposed to work. We shall encounter this 
 supposed law, and modifications of it, again and again. It reap- 
 peared in the work of the earliest English actuary who, so far 
 as we know, was not acquainted with de Witt's work at all. De 
 Witt also foresaw that there would be a selection of the best lives 
 for annuitants, and that a mortality much lower than in the gen- 
 eral population was to be expected. Thus he foresaw a truth 
 which it took the British government many years, involving great 
 loss, to discover. 
 
 The next discovery was by the British philosopher, Halle}^, 
 who, in 1693, published in the "Transactions" of the Eoyal So- 
 ciety the result of his investigations into the mortality of the city 
 of Breslau, Germany, which had given out a complete statement 
 of deaths, by ages, for a number of years. Assuming a stationary 
 population, Halley proceeded to construct a mortality table from 
 the records of deaths only. Halley thus produced a complete 
 mortality table showing the diminution of the number living each 
 year out of an original group. By means of this table he also 
 proceeded to give the now common and always simple formulas 
 for computing the values of annuities. The table which he made 
 is the first that was made from population statistics. The develop- 
 ment of actuarial science was in a large degree due to him, de 
 Witt's work lay buried in the archives of Holland and West Fries- 
 land, and was only uncovered within the last fifty years. 
 
20 DEVELOPMENT OF 
 
 I have said that Halley's Breslau table was the earliest that 
 was drawn from actual experience, and this is true so far as a 
 table showing the mortality at each year of age is concerned. But, 
 as early as 1662, John Graunt had published an analysis of the 
 mortuary returns of London, and had formulated a mortality table 
 in a rough form, but not giving the rate of mortality at each year 
 of age. His work, however, had attracted much attention, and 
 passed through several editions. 
 
 The next great advance was when De Moivre published in 1725 
 his great work on life annuities. A French exile, living in London,, 
 he had already distinguished himself in mathematics, and es- 
 pecially in his works on probabilities. Now he proceeded to apply 
 his learning to the practical purpose of determining the values- 
 of leases for life and other life annuities. Already the summa- 
 tion of series, when following some mathematical law, had been 
 developed to a point which, he plainly saw, rendered it compara- 
 tively simple and easy to compute annuity values, whether for one 
 or for more lives, if only a mortality table could be brought to 
 follow a mathematical law. He investigated the Breslau table 
 put forth by Halley, and discovered that within reasonable limits, 
 for a long period of life at least, and possibly for the whole, the 
 number surviving to each year of age out of a given number set- 
 ting out from the earliest age of the table, was a term in a de- 
 creasing arithmetical series. That is, he found that if the number 
 surviving be treated as a term of such a series, it would diminish 
 very nearly as in the original table. This means that the num- 
 ber of deaths could be assumed to be the same for each year of age 
 out of the number originally setting out. Thus, if the deaths, 
 for instance, were 1,000 per annum out of an original 100,000,. 
 setting out at age 20, this would mean 1,000 out of 100,000 at 
 age 20, 1,000 out of 99,000 at age 21, etc. So the probabili- 
 ties of death would each year increase, not because of any change 
 in the numerator of the fraction, but solely because of the steady 
 and regular diminution of the denominator. Without going into- 
 details at this time, it must be readily seen that for the purpose 
 of determining any value or function whatever, the arrangement 
 of a mortality table as a mere regularly decreasing arithmetical 
 series offered great advantages. 
 
INSURANCE MATHEMATICS 21 
 
 De Moivre also discovered, somewhat later in his career, that 
 complete tables of annuities could be made by a continuous proc- 
 ess, starting from the extreme limit of life, without computing 
 all the values separately which go to make up the value of each 
 annuity. This discovery applied, first, to annuities certain; but 
 its application to life annuities was soon discerned as well. A 
 word of explanation may be needed to make the matter clear to 
 you, and I linger to give it because this is the first great labor- 
 saving invention in the science, which now embraces so many. 
 
 Suppose, then, that you know the value of one, due at the end 
 of one year, and desire to discover the value of an annuity of one 
 for two years. This is plainly worth at the end of one year, and 
 before the first annuity payment is made, one plus the value of 
 one, due at the end of one year. Therefore, at the beginning 
 of the first year it will be worth one plus the present value of 
 one, due at the end of one year, the whole multiplied by the pres- 
 ent value of one, due at the end of one year. And so, by a con- 
 tinuation of this process, a complete table of the values of an- 
 nuities for each of long periods may be successively computed. 
 The same principle, the discounts merely being made also to in- 
 clude the probability as well as the interest factor, applies to life 
 annuities; and this was De Moivre's discovery. It was also inde- 
 pendently seen by both the great mathematician Euler and also 
 by Thomas Simpson, De Moivre's younger rival in England. 
 
 It was in 1742 that the first work of Thomas Simpson upon 
 the subject, his "Doctrine of Annuities and Reversions," made its 
 appearance. Simpson, as was stated in the previous lecture, was 
 a self-taught, non-academic mathematician, believed to be the 
 greatest of such that England has produced. He had but emerged 
 from obscurity and ill fame, owing to conjuring and necromancy. 
 That he should come forth as a rival of De Moivre was almost 
 insupportable by the latter, who was now in the flower of his 
 age and at the zenith of a well-earned reputation. Simpson's 
 book, too, was an attempt to popularize the science, never a popu- 
 lar movement among those who hope to profit by keeping it ex- 
 clusive. And, worst of all, he had little respect for the doctrine 
 of equal decrements, and, indeed, did not spare it. 
 
 Simpson urged a return to Halley's method of taking the facts 
 
22 DEVELOPMENT OF . 
 
 as one found them, instead of seeking to compel the mortality- 
 table to follow a rule to which it was indifferently suited. At the 
 same time, on the basis of De Moivre's principle, he discovered 
 an equal age formula for joint life computations, which greatly- 
 simplified the operations, and which bears his name to this day. 
 But perhaps his greatest distinction is this, that he, first of all 
 actuaries, fully realized the importance of the life insurance phase. 
 Up to his day life insurance was known in but two forms, one 
 being policies for short terms, without privilege of renewal, and 
 the other, insurance for the whole of life for indeterminate 
 amounts, to be determined by the proceeds of assessments. Simp- 
 son introduced the idea of whole life insurance for level annual 
 premiums and, as we shall see, he also afterward introduced the 
 practice. 
 
 At about the same time Des Parcieux, the elder, produced in 
 France his work, "Probabilities of Human Life,'^ published in 
 1746. This was practically the beginning of annuities in that 
 country. It was followed in 1749 by Buffon's "Tables of Prob- 
 abilities of Human Life," which was mainly a discussion of tables 
 already published, but included also some new tables. Des Par- 
 cieux, in his work, gave tables which he had constructed from 
 the experience of two tontine funds and from the mortality in 
 fourteen monasteries. The last became the standard table for 
 insurance in France and remained standard until within the last 
 generation. 
 
 In 1753 the scientist Kersseboom of The Hague brought out 
 a volume discussing the tables of Halley and Buffon, and develop- 
 ing some of the principles of the science. 
 
 James Dodson, an English university man of high standing, 
 began in 1748 the preparation of a volume entitled "Analytical 
 Solution of Problems Eelating to Annuities." Mr. Dodson began 
 to take an interest in the subject comparatively late in his busy 
 life, and his interest was rather sharpened than otherwise by the 
 discovery that, while he might procure life insurance at very high 
 rates in a stock company, and for a very short period, he could 
 get no whole life insurance at all, even in the Amicable on the 
 assessment plan, because he was more than 45 years of age. He 
 became one of the earliest and most active and influential of 
 
INSURANCE MATHEMATICS 23 
 
 Simpson's converts to the view that life insurance could be fur- 
 nished on a whole life plan with equal annual premiums, and 
 that safe premiums for the same could he computed. Together 
 these men labored for several years to bring enough people to- 
 gether to organize a company on the mutual plan — they had no 
 hope of interesting stockholders in so long and uncertain a venture, 
 even in those speculative days. The result was the "Society for 
 the Equitable Assurance of Lives and Survivorships," which, 
 now generally known as "The Old Equitable," survives to this day. 
 The organization was not without difficulties, and had to be ef- 
 fected as an unincorporated voluntary association, practically a 
 copartnership, because the Parliament refused a charter on the 
 ground that the undertaking was very risky. 
 
 The tables which were employed by the Equitable were based 
 upon a mortality table constructed by Dodson from the mortuary 
 statistics of London. Fortunately the experience proved an ex- 
 aggeration of the deaths which the company experienced, and as. 
 it did much more life insurance than annuities, the result was 
 favorable. Of course it would have been disastrous if the con- 
 trary had proven the case, or if the annuity business had ex- 
 ceeded the life insurances, for the same mortality table was em- 
 ployed for both. 
 
 Neither Thomas Simpson nor James Dodson, though they con- 
 tributed so much to the foundation of the Equitable, and indeed 
 were and are recognized to have been its founders, having stirred 
 up interest in the matter by their publications and by lectures 
 upon the subject, in London and elsewljere, was included in the 
 management of the society; but it set forth upon its career on 
 plans devised and furnished by them, and its success is theirs. 
 
 In the same year that the Equitable began operations there 
 appeared a first edition of a most remarkable contribution to act- 
 uarial science, by Dr. Eichard Price, a Unitarian preacher, who 
 was destined to have much to do, both indirectly and directly, 
 with the development of life insurance, and of the Equitable in 
 particular. The work was entitled "Observations on Reversion- 
 ary Payments." It passed into four editions, each much en- 
 larged, during the succeeding twenty-one years, and was the au- 
 thoritative work upon the subject for a very long time, a revised 
 edition being published so late as 1806. 
 
24 DEVELOPMENT OF 
 
 Dr. Price warmly supported Thomas Simpson's view that it 
 ^as folly to construct mortality tables by purely mathematical 
 principles, and urged that investigations of the actual death rates 
 be made as the only means of making a reliable table of mortality. 
 He also encouraged with enthusiasm the work of the Equitable, 
 and in his later editions contrasted sharply its scientific and safe 
 system, assuring solvency, with the unsafe plans and insolvent 
 condition of all the assessment annuity concerns with which the 
 country was then pestered. In addition to his demonstrations of 
 these things he himself analyzed several bodies of population 
 statistics, such as those of Sweden, which had recently been pub- 
 lished, constructing mortality tables from them; and, which proved 
 most important of all, he collected the data concerning deaths m 
 several parishes of Great Britain, notably Northampton, and made 
 mortality tables from these. The Northampton table enjoyed 
 great repute, and was the standard for both life insurance and 
 >)nnuity computations for many years. It was adopted by the 
 Equitable, and Dr. Price was invited to prepare monetary tables 
 on that basis for that company. Later, his nephew, William Mor- 
 gan, was made actuary of the company, a post which he occupied 
 throughout his long life. The theories of De Moivre were, for 
 the time, completely lost sight of, and it was conceded everywhere 
 that the only correct system of developing a mortality table was 
 to follow the facts of statistics closely, not expecting the same 
 to express a mathematical law. William Morgan also distinguished 
 himself by the publication of many books and pamphlets, includ- 
 ing contributions to the Eoyal Society, of which he was an hon- 
 ored member. His most important work, perhaps, was entitled 
 "Doctrine of Annuities and Assurances on Lives and Survivor- 
 ships," published in 1779. To survivorships he devoted much 
 study. He also edited and revised the edition of Eichard Price's 
 "Observations on Keversionary Payments," which appeared in 
 1806. 
 
 In France progress was also being made, although there annui- 
 ties proved more popular by far than life assurances, which is 
 true even to this day. In 1767 the great mathematician, Euler, 
 published a work upon "General Researches into the Mortality 
 and Increase of the Human Race," and in 1781 DesParcieux, the 
 
INSURANCE MATHEMATICS 25 
 
 younger, contributed a "Treatise on Annuities." In Great Britain, 
 Maseres, with a foreign-sounding name, published a large book, 
 of no very great value, entitled "Principles of the Doctrine of 
 Annuities," in 1783; in it he endeavored to make the whole thing 
 a matter of arithmetic. 
 
 There was great progress in life insurance during the last part 
 of the eighteenth century, but little was done in the direction 
 of increasing either the material upon which actuarial science 
 works, or the methods of doing the work. The Northampton table 
 was now fully accepted in Great Britain as the standard, and 
 the government even sold annuities based upon its accuracy. 
 
 In 1810 the monumental work of Francis Bailly upon "Life 
 Annuities and Assurances" appeared. It was the first compre- 
 hensive and thorough treatise that had ever been published upon 
 the subject. What was known had by this time clarified in men^s 
 minds, and it was now possible to treat the topic thoroughly and 
 well. Bailly's equipment for this task was nearly as good as 
 that of La Place for dealing with probabilities; and it is note- 
 worthy that the two epoch-making works made their appearance 
 at about the same time. La Place's "Probabilities" appearing m 
 1812. Bailly's book was so good and so comprehensive that as 
 late as 1864, when by reason of the invention of commutation 
 columns as a convenience for calculations and of a change in the 
 symbols, the book had become antiquated, its worth caused it to 
 be rewritten, edited and produced with modern symbols, so as to 
 continue to be useful to the student. Yet, notwithstanding his 
 abilities and accomplishments, some of which, of course, em- 
 braced the correction of others' errors, or perhaps, more accurately 
 stated, because of these things, Bailly was anything but welcomed 
 by Morgan, now aging, and now also the absolute master of things 
 actuarial. In consequence a feud, now disguised, now breaking 
 forth, raged between the men during their joint career. 
 
 In 1815 appeared Joshua Milne's ^^aluation of Annuities and 
 Assurances," also an epoch-making work ; for, in addition to much 
 that was new and clearer in the text, the author included a new 
 mortality table, drawn from the experience of the parish of Car- 
 lisle. This table was the first to be graduated so that it seemed 
 to express a regular force of mortality, although for the same no 
 
26 DEVELOPMENT OF 
 
 mathematical expression was known or sought. The influence of 
 Eichard Price was yet too powerful to permit turning one's atten- 
 tion in that direction anew. The graduation was by what is now 
 known as the graphic method, and it was necessary, particularly in 
 this case, because the number of lives dealt with was very small. 
 This table showed a much lower mortality rate than the North- 
 ampton table and came far nearer, also, to expressing what the 
 companies were finding to be their own experience. This was 
 even confirmed by the experience of the Equitable itself. In con- 
 sequence most of the younger companies promptly adopted the 
 new table, especially as the Northampton was not merely not 
 fitted for annuities but was positively dangerous. But the North- 
 ampton held its own with the Equitable and some of the older 
 companies, and there was a bitter fight about the matter, the 
 echoes of which have, even now, scarcely ceased to rumble. 
 
 In 1820 appeared Hendry's work on "Life Annuities," and in 
 1825 Griffith Davies' remarkable book, entitled "Tables of Life 
 Contingencies." This work introduced into general use the second 
 great labor-saving device of actuarial science, the commutation 
 column. A suggestion of something of this sort had been made 
 by William Dale as early as 1772 in his "Introduction to the Study 
 of Annuities," and it had also been suggested by William Morgan 
 in the introduction to his principal work, published in 1779. A 
 much more completely developed system had also been devised by 
 the German scientist, Tetens, and published in 1785; but his 
 system was not known to actuaries until much later, having been 
 buried in a work upon another subject. In 1812 Francis Bailly 
 presented to the Eoyal Society a paper by George Barrett, a self- 
 taught English mathematician, in which a system was outlined. 
 The paper was read, but not published, by the Royal Society, an 
 omission which does it little credit, which was attributed to William 
 Morgan, who was upon the committee, and is supposed to have 
 been actuated by envy of Bailly, and in any event did not see the 
 importance of what he had himself suggested, and which called 
 forth the bitterest criticism from Bailly, who included the paper 
 in the second and subsequent editions of his own book. Unfortu- 
 nately, as a result of the indifference of the Royal Society, Barrett 
 was unable to print the tables which he had prepared, and his 
 invention for the time was of no practical use. 
 
INSURANCE MATHEMATICS 27 
 
 Griffith Davies, himself a self-taught mathematician, who had 
 risen by hard work and his talents to a responsible post as actu- 
 ary of a leading London company, developed the idea along some^ 
 what different lines from those employed by any of his prede- 
 cessors, and by adding actual commutation tables, according to the 
 Carlisle table and various rates of interest, he also made his ideas 
 of practical use forthwith. 
 
 In the same year Benjamin Gompertz, a great mathematician, 
 who otherwise gave little attention to actuarial problems, wrote 
 a letter to Francis Bailly concerning a mathematical force of 
 mortality which, according to his view, could be given expression 
 in terms of the infinitesimal calculus and algebra, and which he 
 believed to be at the foundation of all mortality tables. In his 
 letter he explained that the natural law, which he considered to 
 be expressing itself in the mathematical law, was that as men 
 grow older the vitality wanes, and that the measure of this change 
 caused the change in the death rate and could be evolved out 
 by differential methods. Thus within the first twenty-five years 
 of the new century the supremacy of the Northampton table, 
 which at the beginning of that century seemed so securely fixed 
 in that position, was threatened by two foes, one the Carlisle table, 
 which disputed its data, and one Gompertz's discovery, which re- 
 newed the interest in De Moivre's original theory of a mathemat- 
 ical law for mortality, which theory had been regarded as hope- 
 lessly dead. But while the Carlisle table was making its way 
 rapidly, the theory of Gompertz, being found to square with the 
 facts of none of the tables, was for a time thrust aside. 
 
 In 1826 the eminent British mathematician, Charles Babbage, 
 published a work entitled "A Comparative View of Various Life 
 Offices," which showed a marked bias for the Equitable. The book 
 is mainly non-mathematical, however. 
 
 In 1832 T. E. Edmonds published a book in which he further 
 discussed the possibility of employing a mortality table in which, 
 while it accurately represented experience, the graduation was 
 according to a mathematical formula. His book was entitled ^T.ife 
 Tables, Founded upon the Discovery of a Numerical Law Eegu- 
 lating the Existence of Every Human Being, Illustrated by a New 
 Theory of the Causes Producing Health and Longevity." The 
 
28 DEVELOPMENT OF 
 
 system proposed by Mr. Edmonds bears a striking resemblance 
 to that suggested by the originator of actuarial science, John del 
 Witt, whose essay upon the subject was at that time sleeping un- 
 known to all actuaries in the archives of Holland and West Fries- 
 land. Edmonds' system was not adopted, but the ideas which are 
 set forth in the book, and also are plainly indicated by the singu- 
 lar title, have survived, and, indeed, found lodgment in many 
 minds. As Gompertz had thought of the thing, "force of mor- 
 tality," Edmonds thought of the name, and while Edmonds' for- 
 mulas are forgotten, his name for the tendency which he did not 
 accurately measure has become a permanent addition to life in- 
 surance terminology. Already, before Edmonds, Thomas Young 
 espoused the cause of the mathematical formula in a letter which, 
 much to William Morgan's disgust, was accepted and printed by the 
 Eoyal Society in 1826. It was entitled "A Formula for Expressing 
 the Decrement of Life," and it led to a controversy with the 
 nephew of the great Eichard Price, who considered that the repu- 
 tation of himself and his uncle depended upon refuting all argu- 
 ments in favor of a mechanical system. 
 
 The Northampton table was giving way, however, before an 
 enemy more formidable than the mathematical formula; that 
 enemy was experience. For the British government found its 
 experience with annuitants disastrous, having sold annuities at 
 prices fixed by this table; and life insurance companies in general 
 found the table to absurdly exaggerate the losses which they might 
 expect. Consequently the Carlisle table steadily made its way into 
 favor. The publication of the government annuity experience, 
 tabulated by John Finlaison in 1823, and adopted by the govern- 
 ment in 1829, completely displaced the Northampton table as to 
 its use for annuities, and, indeed, it was thereafter retained 
 by no company of importance excepting the Equitable, which stub- 
 bornly retains it, even to this day, or at least was doing so at a 
 recent date. 
 
 Another consideration which occurred to de Witt at the very 
 beginning forced itself upon men's notice in connection with an- 
 nuitants, and that was that there is a very great measure of self- 
 selection among annuitants, assuring that the average of longevity 
 will be much higher than among the population, and also than 
 
INSURANCE MATHEMATICS 29 
 
 among insured lives, in spite of the careful medical selection which 
 companies had learned to make. This made all tables that are 
 suitable for insured lives unsuitable, for that very reason, to 
 measure the value of annuities. The fact was recognized in France 
 before it was in Great Britain, for in France the Duvillard table, 
 named after its author, was introduced so early as 180G, and being 
 better suited to the purpose than DesParcieux's table, was adopted 
 for annuity purposes. In Great Britain the government experi- 
 ence table of 1829 was employed until 1860, when a second gov- 
 ernment table appeared, and this in succession until 1883, when a 
 third government table appeared, all the production of the Fin- 
 laisons, father and son. At this time an unadjusted table has just 
 appeared, drawn from the joint experience with annuitants of the 
 chief British companies; and while it confirms in a remarkable 
 manner the lessons of the government tables, it shows even a 
 greater longevity and will doubtless be mainly employed here- 
 after. 
 
 In Edmonds' work, in addition to discussing the force of mor- 
 tality, he considered the force of sickness, i. e., the liability to 
 disablement, which he deemed to be subject to the same law, be- 
 ing only another expression of it. There appeared in 1835 a work 
 entitled "A Treatise on Friendly Societies,'*' by Charles Ansell, 
 which contained the experience of the Highland Society in rela- 
 tion to sickness, and also the combined experience of a number 
 of Scottish friendly societies. Ansell supplied monetary tables 
 for the same. These were adopted as standard by the friendly 
 societies generally, and as they proved to be too low, much em- 
 barrassment was caused thereby. Xo trace of evidence that Ed- 
 monds' theory was correct was discernible, however, in this; nor, 
 indeed, could its correctness be tested before there were several 
 experiences at hand to compare. The publication of Henry Rat- 
 cliff's report upon the experience of the Manchester Unity of Odd 
 Fellows for 1846-8, and of the report upon the experience of the 
 Foresters, by Frederick G. P. Neison, at about the same time, 
 opened the way for a thorough development of the subject. Two 
 later reports of the Manchester Unity appeared during Mr. Rat- 
 cliff's lifetime, viz., one covering 1856-60, and the other 1866-70; 
 and, as a result of careful comparison, it was developed that, so 
 
30 DEVELOPMENT OF 
 
 far as could be determined a priori, the liability to illness could 
 be represented as a function of the force of mortality. The very 
 recent investigation of the entire aggregate experience of the 
 British friendly societies, during the present year, confirms also 
 the theory that from a mortality table the liability to sickness may 
 be ascertained with much accuracy. It is still, however, a matter of 
 ratio, and the algebraic expression for a force of sickness has not as 
 yet been developed. Yet another report upon later experience of 
 the Manchester Unity is now in preparation. 
 
 With the advent of the commutation method of making com- 
 putations there was naturally a great change in symbols, and a 
 new notation began to come in. In consequence of this, and of 
 the new formulas as well, that were found necessary, the old text- 
 books were rapidly becoming antiquated. Naturally, therefore. 
 Prof, de Morgan's "An Essay on Probabilities and on Their Appli- 
 cation to Life Contingencies and Insurance Offices," published in 
 1838, and especially "A Treatise on the Value of Annuities and 
 Eeversionary Payments," by David Jones, published in 1840, were 
 generally welcomed. The former remained for a long time the 
 best English text-book on probabilities, though not so highly es- 
 teemed as to its actuarial content, while the latter was for more 
 than a quarter century the book of books for the actuarial stu- 
 dent, and is valued unto this day. 
 
 The years from 1840 to 1860 were fruitful in the elucidation 
 of many questions which were very difficult to handle before the 
 commutation system was introduced. They also saw the begin- 
 ning of the great work of England's registrar-general. Dr. Will- 
 iam Farr, author of English Life Tables Nos. 1, 2 and 3, the first 
 two of which appeared during this period. Toward the close of 
 the period the Institute of Actuaries was organized in London, 
 and although this body was frowned upon by many of the actua- 
 ries of great reputation in that day, who refused to join, and or- 
 ganized the Actuaries' Club as a more exclusive and aristocratic 
 society, the Institute, by setting its standard higher and higher, 
 and also by excluding all considerations as to candidates except 
 those of fitness, won a pre-eminent place for itself and also great 
 respect for the entire profession. It also founded a journal in 
 which from that time forward the most valuable discoveries and 
 
INSURANCE MATHEMATICS 31 
 
 inventions have usually been first given to the world. The his- 
 tory of the progress of actuarial science may be read since its 
 foundation in the volumes of this journal. 
 
 To this "Journal of the Institute of Actuaries" William Make- 
 ham, in 1860, contributed what has proved to be a solution of the 
 disagreement between the school that followed De Moivre and 
 the school that followed Dr. Price. In other words, he discovered 
 and gave to the profession the extension of the principle laid 
 down by Gompertz, which was needed to reconcile his theory as 
 to the force of mortality with the facts of experience. Gompertz 
 had guessed that increasing mortality was the consequence of wan- 
 ing vitality, and could be measured by a steadily increasing func- 
 tion. Makeham divined that deaths might be divided into two 
 classes, viz., from causes that apply with almost equal force at 
 all ages, such as accidents, and from causes which flow from di- 
 minished vitality. A modification of the formula of Gompertz 
 to agree with this idea gives an expression for the force of mor- 
 tality which has proved most useful, and also to measure with great 
 accuracy the facts of experience. 
 
 Notwithstanding this discovery, so strong was the force of tra- 
 dition and the influence of the school of Dr. Price that when a 
 table of mortality was constructed from the experience of 20 
 British companies, under the auspices of the Institute, the work 
 being completed and published in 1869, the same was graduated 
 by a formula which was really merely a very smooth mathematical 
 expression for the graphic method; and it was not before 1887 
 when the second volume of the "Institute of Actuaries' Text-book*' 
 was published, that tables were published in Great Britain which 
 were graduated by Makeham's formula. It was found that such 
 graduation gave results very close to the original data, and the 
 system is now practically universally accepted. It presents all 
 the advantages in the matter of facility of computing joint life 
 functions that were originally foreseen by Simpson and others. 
 
 The 20 British Offices table was not the first combined expe- 
 rience table to be constructed in Great Britain. In 1843 a table, 
 known in Great Britain as the 17 Offices, and in America as the 
 Actuaries', or Combined Experience Table, was published, having 
 been imder construction for a number of years. It was hotly 
 
32 DEVELOPMENT OF 
 
 attacked in England by both the friends of the Carlisle table and 
 the lingering champions of the Northampton table, and it never 
 became standard there; though tables of monetary values were 
 published by Jenkin Jones. 
 
 In the United States, on the contrary, this table, chancing to 
 have come out at about the right moment, and appealing to Elizur 
 Wright as a fair test of solvency, was introduced by him in 1859 
 as a standard for valuing the policies of companies doing busi- 
 ness in Massachusetts, of which State he was commissioner, in 
 consequence this table was pretty widely adopted by American 
 companies and departments, and is yet in very common use. 
 
 Early in the sixties there was. published a table drawn from the 
 experience of the Mutual Life Insurance Company. This table 
 was graduated by Sheppard Homans, actuary of that company. 
 A few years later Mr. Homans published another table embracing 
 a modification of the Actuaries' table, by the experience of the 
 Mutual Life, and with some other changes. This table was given 
 the name American Experience Table, and was adopted as the 
 standard in New York and in other States, as well as by the Mutual 
 Life and many companies. It is now also in common use. 
 
 As in Great Britain, so also in the United States, the first 
 table giving the combined experience of the home companies has 
 been discredited and little used. In 1870 a table covering the 
 experience of 36 American offices, which had been prepared under 
 the direction of the Chamber of Life Assurance and a committee 
 of prominent actuaries, made its appearance. Although it was 
 graduated according to Makeham's formula, and thus afforded 
 many advantages in actuarial operations, and although it was 
 accompanied also by a most elaborate set of monetary tables, it 
 was adopted by no company of consequence, and has never been 
 standard anywhere. 
 
 The principal contributions of America to actuarial mathe- 
 matics have been three formulas for the computation of individual 
 reserves by what is known as the retrospective method, which are 
 known by the names of their authors, as follows : Wright's, Fack- 
 ler's and McClintock's formulas. The first two are more com- 
 monly employed to compute values of policies involving one life 
 only, and the last to compute the values of joint life policies, 
 
INSURANCE MATHEMATICS 33 
 
 for which purpose it is singularly useful. Another important 
 discovery was the contribution plan of dividing surplus, the honor 
 of which is due to Sheppard Homans and David Parks Fackler, 
 jointly. 
 
 The most recent developments in the mathematics of insur- 
 ance are due to one man, who has by competent authority been 
 pronounced the greatest living actuary. Dr. Thomas Bond Sprague. 
 They are the discovery that a select mortality table, instead of 
 the mortality tables commonly in use, represents most closely the 
 iacts of experience in life insurance companies. Thus the usual 
 form of mortality table exhibits the average death rate at a cer- 
 tain age, for instance, without regard to whether the lives are 
 freshly selected by medical examination or not. It has long been 
 understood that the mortality during early years of insurance is 
 comparatively light, and the theory of adverse selection by dis- 
 continuances had been exploited long before Dr. Sprague's time. 
 Indeed, in the analysis which actuaries made of the mortuary ex- 
 perience of the Equitable Society when first published, they clear- 
 ly distinguished the effects of adverse selection for twenty years 
 after entry. But Dr. Sprague discovered that, in the later expe- 
 rience of companies, embodied in the data for the 20 Offices' table 
 the effects of favorable selection appeared to wear off in about 
 five years. So he recommended the use of select tables, com- 
 posed of the actual experience during the first five years of in- 
 surance, combined with the general experience upon lives, admitted 
 more than five years. He prepared and published the monetary 
 tables necessary to give his views effect. 
 
 This was followed some years later by another discovery and 
 suggestion of even greater practical importance, viz., that the 
 net reserve system, as usually employed, was inaccurate, and should 
 be modified. Thus he discovered that, owing to the circumstance 
 that expenses are much higher the first year of insurance than 
 later, the reserve which the usual methods brought out was neither 
 just according to the fundamental meaning of the prospective 
 nor of the retrospective formulas, because it would not be needed 
 to make good the deficiencies of future premiums, the original 
 premiums being adequate for all purposes; nor could it have been 
 accumulated from past premiums, the conditions being as per 
 
34 DEVELOPMENT OF INSURANCE MATHEMATICS 
 
 assumption, and expenses being incident, as they actually arc. 
 He presented his views before British actuaries at various times, 
 and at length before the first Congress of Actuaries in 1895. The 
 issues which he raised are now being fought out by American 
 companies and departments, mainly in the courts, in disputes as 
 to construction of contracts, but also between opposing schools of 
 actuaries. In France and some other countries the principles 
 laid down by Dr. Sprague have already been adopted. 
 
 In the foregoing, upon looking it over, I find that I have 
 forgotten to mention the invention of a system of conversion of 
 annuities to insurknce, and vice versa, and the production of a 
 most remarkable series of conversion tables for annuities and life 
 insurances, which was printed in 1856. It was the work of Will- 
 iam Orchard, one of the most brilliant young mathematicians of 
 Great Britain, whose early death was justly deplored. I have also 
 forgotten to mention the work of David Chisholm, whose series 
 of tables have been most valuable aids to actuaries. 
 
 There has been no lack of text-books in the last half of the 
 nineteenth century. The Institute of Actuaries followed up its 
 great work of collecting the combined mortality experience of 
 twenty companies into a mortality table by putting forth a text- 
 book which is now everywhere the standard authority for higher 
 education upon the subject. It was published in two volumes, the 
 first dealing with annuities certain and the second dealing with 
 life contingencies. A second edition of the first volume is now 
 about to appear, and a second edition of the second volume is 
 m preparation. In America there have also been published text- 
 books, though none which even pretends to the thoroughness of 
 the Institute of Actuarh 
 
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