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\begin{center}
  {\Large DE WITT'S} \\
  \vspace{5cm}
  {\Huge \textbf{Treatise on Life Annuities}} \\
  \vspace{5cm}
  IN A SERIES OF \\
  \vspace{5cm}
  {\Large LETTERS TO THE STATES-GENERAL.}
\end{center}

\pagebreak

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\begin{center}
  DE WITT'S \\
  \medskip
  Treatise on Life Annuities \\
  \medskip
  IN A SERIES OF \\
  \medskip
  LETTERS TO THE STATES-GENERAL.\footnote{The title of the Treatise in the
original (now in the State Archives at the Hague) is ``\textit{Waardye van 
lyf-renten naer proportie van Losrenten,}'' which Mr.\ Hendriks has translated
into English\hfill\textsc{R. G. B.}}                               
\end{center}

\begin{flushleft}
  ``NOBLE AND MIGHTY LORDS:
\end{flushleft}

\noindent
``IN so extensive an administration as that of the united country of 
Holland and West Friesland, it is better,as I have several times stated 
to your Lordships, for several reasons perfectly well known to you, to 
negotiate funds by life annuities, which from their nature are infallibly 
terminable, than to obtain them at interest, which is perpetual, or by 
redeemable annuities; and that it is likewise more useful for private 
families, who understand economy well, and know how to make a good 
employment of their surplus in augmenting their capital, to improve 
their money by life annuities, than to invest it in redeemable annuities 
or, at interest at the rate of 4 per cent.\ per annum; because the above 
mentioned life-annuities, which are sold even at the present time at 14 
years' purchase, pay, in fact, much more in proportion than redeemable 
annuities at 25 years' purchase.  I have consequently respectfully to 
submit to your Lordships the unchallengeable proof of my assertion, and 
at the same time to respond to the wish manifested by the members of this 
body to have such a proof in writing.  That proof, founded on a solid basis, 
is proposed to your High Mightinesses in the following manner:--

``Value of Life Annuities in Proportion to Redeemable Annuities.

``I lay down the following presupposition, in order to determine the 
proportion of a life annuity to a redeemable annuity.  For example, in 
presupposing that the redeemable annuity is and will be current at 25 
years' purchase, or at the rate of 4 per cent.\ per annum, we must find 
at how many years' purchase the life annuity should be sold, to be in 
proportion to the aforesaid redeemable annuity, in such manner that the 
life annuity may, if not with mathematical precision, at least in its 
discovered value, be more advantageous to the purchaser than an annuity 
redeemable with the same capital.

\begin{center}
  ``FIRST PRESUPPOSITION
\end{center}

``I presuppose that the real value of certain expectations or chances of 
objects, of different values, must be estimated by that which we can 
obtain from equal expectations or chances, dependent on one or several 
equal contracts.  Let us take, for example, a small matter, and under 
circumstances intelligible at first sight:--- A person has 2 different 
expectations or chances which may easily lead, the one to nothing, the 
other to 20 stuyvers.  If, by one or several equal contracts, he can 
obtain for 10 stuyvers 2 like expectations, we must estimate that the 2 
aforesaid chances are worth to him exactly 10 stuyvers, because he can 
really obtain for 10 stuyvers these 2 expectations or chances, by making 
an agreement with another person that each of them should stake 10 stuyvers, 
and then gamble or draw lots, by odd or even, head or tail, blank or prize, 
or in some such way, to determine which of the two should have the 20 
stuyvers; thus by the said contract, equal in every regard, he evidently 
finds himself in the position of having in reality the 2 expectations 
or chances, the one of nothing, the other of 20 stuyvers.

\begin{center}
  ``SECOND PRESUPPOSITION
\end{center}

``That in taking at pleasure some years of a man's life, limited to the 
time when he is in full vigour, and neither too young, nor too advanced 
in age; (this space of years shall be here 50 years, namely from the third 
or fourth year of his age, up to the fifty-third or fifty-fourth year;) 
it is not more likely that this man should die in the first half-year of 
a given year, than in the second year; similarly, it is not more likely 
that he should die in the second half-year of the aforesaid year than in 
the first half.  But although it depends entirely on chance whether this 
man, after having lived to the given year, and dying in the course of that 
year, should demise in its first or second half, one finds nevertheless 
in this regard an equality of likelihood or chance similar to the case of 
a tossed penny, where there is an absolute equality of likelihood or chance 
that it will fall head or tail, although it depends entirely upon chances 
as to the side on which it shall turn, and this to so high a degree that 
the penny may fall head 10, 20, or more times following, without once 
calling tail; and vice versa.

\begin{center}
  ``THIRD PRESUPPOSITION
\end{center}

``That a man having passed the aforesaid vigorous time of his life, namely 
the fifty-third or fifty-fourth year of his age, it begins to be more likely 
that he should die in a given year or half-year of the second period that 
has previously been the case; or that it is not more likely with respect 
to another man of like constitution or state of body, that the latter 
should die in less than a year or half-year of the said vigorous time of 
his life; whilst this likelihood or chance of dying in a given year or 
half-year of the first 10 following years, namely from 53 to 63 years of 
his age, taken inclusively, does not exceed more than in the proportion 
of 3 to 2 the likelihood or chance of dying in a given year or half-year 
during the aforesaid vigorous period of life: so that, taking for example 
two persons of equal constitution, one aged 40 years, and the other aged 
58 years, if these two persons made such a contract, that in case the 
person of 58 years should happen to die in less than six months, the one 
aged 40 were to inherit a sum of 2,000 florins from the property of the 
defunct; but that if, on the other hand, the person aged 40 years should 
die in less than six months, the one aged 58 years here to have 3,000 
florins from the property of the deceased; such a contract cannot be 
considered disadvantageous for the person who would have the 3,000 florins, 
if the event were favourable to him, and who, in the contrary event, 
would only lose 2,000 florins.

``I then presume that the greatest likelihood of dying in a given year or 
half-year of the second series of the ten following years (that is, from 
63 years to 73, taken one with the other, rather than in a given year or 
half-year of the period of the vigour of life) cannot be estimated at more 
than double, or as 2 is to 1; and as the triple, or as 3 is to 1, during 
the 7 following years, that is, from 73 years to 80.

``Finally, in supposing that life necessarily ends at the twenty-seventh 
year after the expiration of 50 years of age above presumed, this time is 
neither assumed too high, not too low a standard, as experience manifestly 
teaches us that the life of some men exceeds by a considerable period the 
age of 80 years, the age of 81 years, and even more.

``These three at articles being presupposed, we have, by a demonstrative 
calculation, mathematically discovered and proved that the redeemable 
annuity being fixed at 25 years' purchase, as above, the life annuity 
should be sold at 16 years' purchase, and even higher, to be in equality, 
one with the other; so that in the purchase of 1 florin of life annuity, 
on a young and vigorous nominee, more than 16 florins should be paid, as 
is proved by the following demonstration:--

\begin{center}
  ``FIRST PROPOSITION
\end{center}

``The value of several equal expectations or chances, a certain sum of 
money or other objects of value pertaining to chance, is found to be exactly 
determined by adding the money or other objects of value represented by 
the chances, and by then dividing the sum of this addition by the number 
of chances: the quotient or result indicates with precision the value of 
all these chances.

``To give greater clearness to the demonstration, let a person named 
John have, for example, 3 equal expectations or chances---one of a certain 
pearl, or of 2,000 florins; the second of a certain ruby, or of 3,000 
florins; and the third of a certain diamond, or of 4,000 florins; as beneath.

\begin{center}
  \begin{tabular}{cllrc}
    \multicolumn{2}{l}{\hspace{-0.7cm}Chances.}      \\
    1 & pearl,   & or & 2000 & florins. \\
    1 & ruby,    & or & 3000 &   ``     \\
    1 & diamond, & or & 4000 &   ``     \\
    \cline{1-1}\cline{4-4}
    3 &          &  3)& 9000 &    ``    \\
             \cline{4-4}
      &          &    & 3000 & florins.
  \end{tabular}
\end{center}

\noindent
I say that the 3 above-mentioned expectations or chances are together 
worth to him precisely the third of the above-mentioned objects or sums 
of money, first added up, and then divided by 3, which is the number of 
chances.

\begin{center}
  ``DEMONSTRATION
\end{center}

``In the first place, let John have purchased, in community with two other 
persons, namely, Peter and Paul, and let each of the two have paid 
one-third of the value of the 3 jewels before mentioned; or rather that 
John with Peter and Paul have made common purse, by each contributing 
3,000 florins, which has evidently been an equal contract.

``In the second place, let John have agreed to cease his communityship 
with his two partners, or for other reasons to draw lots by three tickets, 
namely, two tickets blank and one ticket prize, for the 3 above-named 
jewels, or the aforesaid common purse of 9,000 florins of capital, so that 
each of them may draw one of the aforesaid tickets, and that fortune may 
thus point out to which of them she assigns the above-named jewels or 
the whole purse; which is again evidently an equal contract.

``In the third place, let John have agreed with Peter in particular, 
that if fortune favors one of them, in drawing the 3 aforesaid jewels or 
the whole purse, the winner should give the loser the pearl, or 2,000 
florins out of the purse; which is likewise evidently an equal contract.

``In the fourth place, let John have agreed with Paul in particular, that 
if the jewels or purse should fall by lot to one of the two, the winner 
should in compensation give the loser the ruby, or the 3,000 florins out 
of the purse; which is indisputably an equal contract.

``The four conventions or contracts being thus entered upon, the matter 
as concerns John is reduced to this, that he has 3 easy and equal 
expectations or chances - that is to say, one chance of the pearl, or of 
2,000 florins, if fortune favors Peter, who, in compliance with and in 
virtue of the third above-named contract, made with him in particular, 
must give up to John the pearl, or 2,000 florins; one chance of the ruby, 
or of 3,000 florins, if fortune favor Paul, who, from the tenor of the 
fourth contract made with him in particular, must give John the ruby, 
or 3,000 florins; and, lastly, one chance of the diamond, or of 4,000 
florins, if fortune favor himself, (John) since, by virtue of the two 
aforesaid, particular contracts, John having to hand over to Peter the 
pearl, or 2,000 florins, and to Paul the ruby, or 3,000 florins, yet 
retains for himself the diamond, or 4,000 florins; which chances all 
proceed from the aforesaid jewels, or from the purse of 9,000 florins, 
drawn by lot; so that, because John can obtain the proposed expectations 
by a third share of the 3 jewels, or by a capital of 3,000 florins, such 
third of the 3 aforesaid jewels, or the capital of 3,000 florins, is the 
real value of the expectation or chances proposed in the first 
presupposition.  We will in the same manner demonstrate the proposition 
when there are 2, 4, 5, 6, equal expectations or chances, and even more, 
of objects of different value, provided that we assume in greater or 
lesser value, provided that we assume in greater or lesser proportion 
as many contractors with or partners of John, as also in greater or 
lesser proportions as many particular contracts made with each of his 
partners; therefore, the proposition is generally demonstrated.

\begin{center}
  ``COROLLARY
\end{center}

``From that which precedes, we may easily conclude that the 
before-described rule is not the less decisive, although some of the 
expectations or chances be of zero or nothing; because in such case the 
demonstration requires no further change than to suppose one associate 
or partner more than the number of objects of value, relatively to the 
expectations; and further, that no contract like the above is made with 
the partners or associates.

``If, for example, John has the following expectations or chances, namely,---

\begin{center}
  \begin{tabular}{clcccclrc}
    \multicolumn{2}{l}{\hspace{-0.7cm}Chances.}                             \\
      1 & of zero or nothing   & `` & `` & `` & `` &    &      0 & florins. \\
      1 & of a certain pearl   & `` & `` & `` & `` &    &  2,000 &   ``     \\
      1 & of a certain ruby    & `` & `` & `` & `` &    &  3,000 &   ``     \\
      1 & of a certain diamond & `` & `` & `` & `` &    &  7,000 &   ``     \\
      4 &                      &    &    &    &    & 4) & 12,000 &   ``     \\
      \cline{1-1} \cline{8-8}
        &                      &    &    &    &    &    &  3,000 & florins. \\
                  \cline{8-8}
  \end{tabular}
\end{center}

I say that the four above mentioned chances are together worth to him 
precisely the quarter of the three above named jewels, or the sum of 
3,000 florins; for supposing that John, having bought with Peter, 
Paul and Nicholas, the three aforesaid jewels, or that each of them 
having furnished 3,000 florins, they have made a common purse of 12,000 
florins, and then that he makes a general agreement with them, and with 
Peter and Paul, each separately, but not with Nicholas, a special contract, 
(similar to that made by him above,) the matter as concerns John is reduced 
to this, that he has four equal expectations or chances, namely, one 
chance of zero or nothing, if fortune favor Nicholas, with whom he has 
not entered into a special agreement relative to any reciprocal 
reimbursement;--- one chance of the pearl, or of 2,000 florins, if fortune 
favor Peter, who, in such case, and by virtue of the special contract 
made between the two, has to give up the pearl to him, or make good 
2,000 florins;--- one chance of the ruby, or of 3,000 florins, if fortune 
favor Paul, who, in such case, and according to the special contract, 
must hand over to John the ruby, or 3,000 florins;--- and, lastly, one 
chance of the diamond, or of 7,000 florins, if fortune favor himself, 
(John,) because, by virtue of the above mentioned special contracts 
made with Peter and Paul, to whom respectively he has to hand the pearl, 
or 3,000 florins, and the ruby, or 3,000 florins, he yet retains for 
himself the diamond, or 7,000 florins;--- which chances all proceed from 
the aforesaid jewels, or from the purse of 12,000 florins, drawn by lot.

``And it is to be observed, that I have here expressly made use of an 
example or case of three objects of value, without expression of any sum, 
as in speaking of a pearl, a ruby, or a diamond, so as to cause the 
demonstration to be applicable to all sorts of numbers, to fractions as 
well as integer numbers, to irrational as well as to rational numbers, 
since all imaginable numbers may be applied to the value of these jewels.

\begin{center}
  ``SECOND PROPOSITION
\end{center}

``If any one has different equal expectations or chances, of which some 
will cause him to obtain each a certain sum of money or other object of 
value, and the others will produce him nothing at all; if, besides, he 
possesses several other chances, each of a certain sum of money or object 
of value; and further, if he has some other chances, each of a certain 
sum of money or object of value, and so on;--- we find the actual value of 
the aforesaid chances, by multiplying each item or sum of money, relative 
to each expectation in particular, by the quantity or number of existing 
chances, then adding the products of the resulting multiplications of 
these partial operations: we finally divide the sum, or mass of partial 
products, by the collective number of chances, and the quotient indicates 
exactly the value of all these chances.

``Suppose, for example, that a person has the following chances of the 
objects or value annexed:--

\begin{center}
  \begin{tabular}{ccccccrccccr}
    \multicolumn{4}{l}{\hspace{-0.7cm}Chances.}
     & \multicolumn{5}{c}{Each Chance of} \\
     6 & `` & `` & `` & `` & `` &     0 & `` & `` & `` & `` &      0 \\
     6 & `` & `` & `` & `` & `` & 1,200 & `` & `` & `` & `` &  7,200 \\
     4 & `` & `` & `` & `` & `` & 2,100 & `` & `` & `` & `` &  8,400 \\
     3 & `` & `` & `` & `` & `` & 3,600 & `` & `` & `` & `` & 10,800 \\
     2 & `` & `` & `` & `` & `` & 4,200 & `` & `` & `` & `` &  8,400 \\
     \cline{1-1} \cline{12-12}
    21 &    &    &    &    &    &       &    &    &    &    & 34,800 \\
       &    &    &    & \multicolumn{5}{c}{21)34,800(1,657 1-7}\footnote{In
    the original, the old galley form of division (or, as the Dean of Ely
    terms it, the scratch method of division,) is here employed, and also in
    the subsequent examples.  I have not, however, thought it worth while to
    trouble the printers with the now strange and obsolete type it requires.}
  \end{tabular}
\end{center}

\begin{center}
  ``DEMONSTRATION
\end{center}

\noindent
I say that all the above mentioned chances are together worth to this 
person exactly $1,657\frac{1}{7}$; a value which we find, as is mentioned in 
the proposition, by multiplying each item, namely, 1,200 by 6; 2,100 by 4; 
3,600 by 3; and 4,200 by 2; then adding the products of these multiplications, 
that is to say, 7,200, 8,400, 10,800, and 8,400, and dividing the sum total, 
or 34,800, by 21, which is the collective number of chances.

``Because we can represent the above chances reduced to their unities, 
as well as their values, in the following manner:--

\begin{center}
  \begin{tabular}{l}
      \text{Chances}\hspace{9.2cm}\text{of} \\ \smallskip
      6 $\left\{
        \begin{array}{ccccccr}
         1 & \qquad.\qquad & \qquad.\qquad 
           & \qquad.\qquad & \qquad.\qquad & \qquad.\quad & \phantom{00,00}0 \\
         1 & . & . & . & . & . & 0 \\
         1 & . & . & . & . & . & 0 \\
         1 & . & . & . & . & . & 0 \\
         1 & . & . & . & . & . & 0 \\
         1 & . & . & . & . & . & 0
         \end{array}\right.$ \\
  \end{tabular}
\end{center}
\begin{center}
  \begin{tabular}{l}
      6 $\left\{
        \begin{array}{ccccccr}
         1 & \qquad.\qquad & \qquad.\qquad 
           & \qquad.\qquad & \qquad.\qquad & \qquad.\quad & \phantom{0}1,200 \\
         1 & . & . & . & . & . & 1,200 \\
         1 & . & . & . & . & . & 1,200 \\
         1 & . & . & . & . & . & 1,200 \\
         1 & . & . & . & . & . & 1,200 \\
         1 & . & . & . & . & . & 1,200
         \end{array}\right.$ \\
  \end{tabular}
\end{center}
\begin{center}
  \begin{tabular}{l}
      4 $\left\{
        \begin{array}{ccccccr}
         1 & \qquad.\qquad & \qquad.\qquad 
           & \qquad.\qquad & \qquad.\qquad & \qquad.\quad & \phantom{0}2,100 \\
         1 & . & . & . & . & . & 2,100 \\
         1 & . & . & . & . & . & 2,100 \\
         1 & . & . & . & . & . & 2,100 \\
         \end{array}\right.$ \\
  \end{tabular}
\end{center}
\begin{center}
  \begin{tabular}{l}
      3 $\left\{
        \begin{array}{ccccccr}
         1 & \qquad.\qquad & \qquad.\qquad 
           & \qquad.\qquad & \qquad.\qquad & \qquad.\quad & \phantom{0}3,600 \\
         1 & . & . & . & . & . & 3,600 \\
         1 & . & . & . & . & . & 3,600 \\
         \end{array}\right.$ \\
  \end{tabular}
\end{center}
\begin{center}
  \begin{tabular}{l}
      2 $\left\{
        \begin{array}{ccccccr}
         1 & \qquad.\qquad & \qquad.\qquad 
           & \qquad.\qquad & \qquad.\qquad & \qquad.\quad & \phantom{0}4,200 \\
         1 & . & . & . & . & . & 4,200 \\
         \cline{1-1} \cline{7-7}
         \end{array}\right.$ \\
  \end{tabular}
\end{center}
\begin{center}
  \begin{tabular}{l}
      \phantom{2\{} $
        \begin{array}{ccccccr}
         1 & \qquad\phantom{.}\qquad & \qquad\phantom{.}\qquad 
           & \qquad\phantom{.}\qquad & \qquad\phantom{.}\qquad 
           & \qquad\phantom{.}\quad & 34,800
         \end{array}$ \\
  \end{tabular}
\end{center}
And being so represented, according to the solution in the first proposition, 
their value would be found by adding all the values, i.e.\ zero or 0 six 
times; 1,200 six times, which is the same thing as if the number 1,200 were 
multiplied by 6; 2,100 four times, or as if 2,100 were multiplied by 4; 
3,600 three times, or like that number multiplied by 3; and, lastly, 4,200 
twice, or as though 4,200 were multiplied by 2; then by dividing the sum 
total by the number of chances, namely 21: and as the above summary addition 
of all the items individually treated, evidently does not differ from the 
addition resulting from the multiplication of the items in aggregate, but 
is identical, namely 0; 7,200; 8,400; 10,800; 8,400; (the sum of these 
final items being divided by the same number,- that is, by the sum of the 
collective chances, or 21,) the same quotient must necessarily be obtained, 
which proves that the proposition is true.

\begin{center}
  COROLLARY
\end{center}

``It plainly results from the foregoing proposition, that in a strict sense 
it is not the number of chances of each value which we must consider, in 
the application of the aforesaid rules, but solely their reciprocal 
proportion.  For it is manifest that the divisor augments or diminishes 
by the result of the addition of increase or decrease in chances, in the 
same proportion as the dividend or number to divide increases or 
diminishes by multiplication, when the ratio between the chances 
remains the same; so that, as in the aforesaid case, the divisor and 
dividend remain reciprocally in the same ratio or proportion, and the 
quotient remains unchanged, being the real value of the chances together.  
This will be better understood by the preceding example, treated as 
follows in three different methods, with their solution:--

\begin{center}
  \begin{tabular}{rcccrcccr}
           \multicolumn{9}{c}{I.} \\
           \multicolumn{4}{l}{\hspace{-0.7cm}Chances}&of\quad\ \\
           6 & `` & `` & `` &     0 & `` & `` & `` &      0 \\
           6 & `` & `` & `` & 1,200 & `` & `` & `` &  7,200 \\
           4 & `` & `` & `` & 2,100 & `` & `` & `` &  8,400 \\
           3 & `` & `` & `` & 3,600 & `` & `` & `` & 10,800 \\
           2 & `` & `` & `` & 4,200 & `` & `` & `` &  8,400 \\
           \cline{1-1} \cline{9-9}
          21,& \multicolumn{7}{l}{divisor}         & 34,800 \\
          \multicolumn{9}{c}{$34,800 \div 21 = 1,657\ 1-7$}
  \end{tabular}
\end{center}

\begin{center}
  \begin{tabular}{lcccrcccr}
    \multicolumn{9}{c}{II.} \\
    \multicolumn{4}{l}{\hspace{-0.7cm}Chances}&of\quad\ \\
    1 & `` & `` & `` &                            0 & `` & `` & `` &      0 \\
    1 & `` & `` & `` &                        1,200 & `` & `` & `` &  1,200 \\
    $\phantom{0}\frac{2}{3}$ & `` & `` & `` & 2,100 & `` & `` & `` &  1,400 \\
    $\phantom{0}\frac{1}{2}$ & `` & `` & `` & 3,600 & `` & `` & `` &  1,800 \\
    $\phantom{0}\frac{1}{3}$ & `` & `` & `` & 4,200 & `` & `` & `` &  1,400 \\
    \cline{1-1} \cline{9-9}
    $3\frac{1}{2}$,& \multicolumn{7}{l}{divisor}         & 5,800 \\
    \multicolumn{9}{c}{$5,800 \div 3\frac{1}{2} = 1,657\ 1-7$}
  \end{tabular}
\end{center}

\begin{center}
  \begin{tabular}{rcccrcccr}
           \multicolumn{9}{c}{III.} \\
           \multicolumn{4}{l}{\hspace{-0.7cm}Chances}&of\quad\ \\
           18 & `` & `` & `` &     0 & `` & `` & `` &      0 \\
           18 & `` & `` & `` & 1,200 & `` & `` & `` & 21,600 \\
           12 & `` & `` & `` & 2,100 & `` & `` & `` & 25,200 \\
            9 & `` & `` & `` & 3,600 & `` & `` & `` & 32,400 \\
            6 & `` & `` & `` & 4,200 & `` & `` & `` & 25,200 \\
           \cline{1-1} \cline{9-9}
          63,& \multicolumn{7}{l}{divisor}         & 104,400 \\
          \multicolumn{9}{c}{$104,400 \div 63 = 1,657\ 1-7$}
  \end{tabular}
\end{center}

``From the reasons before mentioned, we obtain in the above three 
examples, by means of the operation of the rule, one and the same quotient 
to determine the total value of all the chances, namely $1,657\frac{1}{7}$  
(It would be the same in every similar case.)

\begin{center}
  THIRD PROPOSITION
\end{center}

``Each half-year of life is equally destructive or mortal to a person 
aged 3 or 4 years, to 53 or 54 years; in such a period he is neither 
too young, nor too aged, to be wanting in the vigour needful for the 
prolongation of his days: so that there is not greater hazard nor likelihood 
that the day of his death should arrive in the first than in the second 
half-year of this vigorous period, and vice-versa; nor that the day of 
his decease should occur rather in these two aforesaid half-years, 
considered each in its individuality, than in the third half-year, and 
vice-versa.  And thus with the other half-years during the aforesaid 
space of time.

\begin{center}
  ``DEMONSTRATION
\end{center}

``Any year of the vigorous period of life of the aforesaid person, being 
taken at pleasure, the first half of that year, or the first six months, 
is as destructive or mortal to him as the second six months.  (According 
to the second proposition.)

``And taking a second or other year of this period of the vigour of his 
life, in setting out from the second half-year of the first year taken, 
which ends consequently just six months after the expiration of that first 
year, the first half of the second year, which thus becomes the second 
half-year of the first year, is quite as destructive or mortal to him as 
the second half of the second year, which is thus the third half-year, 
reckoning as before.  But as the first half-year, as well as the third, 
is as destructive, or mortal as the second, the first half-year and the 
third, compared with each other, are so likewise, since each of them in 
particular is as destructive or mortal as the second half-year; therefore, 
the aforesaid half-years, namely, the first, the second, and the third, 
each separately considered, are equally mortal.

``We might also demonstrate in the same manner that the second half-year 
and the fourth, when the one is compared with the other, are equally 
mortal; and again, that consequently the first half-year, the second, 
the third, and the fourth, each considered by itself, has the same 
chance of destructiveness: it is the same thing for all the preceding 
or subsequent half-years, comprised in the above time of the vigour 
of life;--- which was to be demonstrated.

\begin{center}
  ``COROLLARY
\end{center}

``It results from what precedes, and from the third presupposition, that 
as life annuities are paid in all the offices of Holland and West 
Friesland by half-yearly instalments, or from six months to six months, 
that the annuitant loses all his capital, and receives no return whatever 
from it, if the life upon which the annuity is sunk happen to die in the 
first half-year after the purchase, or do not live six whole months.  
The annuity sunk is supposed to be 1,000,000 of florins, or 20,000,000 
stuyvers, per annum, in order that an exact calculation may be made without 
fractions: therefore, if the above-mentioned life survive a complete 
half-year and do not die in the course of the second half-year, the 
annuitant has then drawn 10,000,000 stuyvers, from which a deduction 
is made of 4 per cent.\ per annum for a half-year, it would have been 
worth to him in ready cash (that is to say, on the day of purchase of 
the said annuity,) 9,805,807 stuyvers, which he would have had to pay, 
if taken at the true value.  If the above life survive so long as two 
complete half-years, and die in the third half-year, the annuitant has 
then drawn 10,000,000 stuyvers after the expiration of the first half-year, 
and after that of the second half-year likewise 10,000,000 stuyvers; 
which sums, deduction being made at 4 per cent.\ per annum, one for a 
half year or six months, and the other for a complete year, would have 
been worth to him in ready cash, or upon the day of purchase of the said 
annuity, 19,421,192 stuyvers, and so on, according as the day of decrease 
were to occur in the fourth, fifth, sixth, or further number of half-years, 
which would have been worth to him each time as many terms or half-yearly 
sums of 10,000,000 stuyvers as complete half-years had elapsed from 
the purchase of the annuity, deduction being made as above of the 
respective discounts.  The computed amounts are specially given in the 
following table:--

\begin{center}
  \begin{supertabular}{cccrrccccr}
     \multicolumn{7}{c}{\textit{If the Nominee survive}} \\
     \multicolumn{7}{c}{\textit{the following Term of Life.}} \\
     \multicolumn{7}{c}{\textit{Half-years.}} & & & \textit{Stuyvers} \\
     \phantom{.} &\phantom{.} & \phantom{.} & \phantom{.} 
           &   0 & . & . & . & . &               0 \\
     & & & &   1 & . & . & . & . &       9,805,807 \\
     & & & &   2 & . & . & . & . &      19,421,192 \\
     & & & &   3 & . & . & . & . &      28,849,853 \\
     & & & &   4 & . & . & . & . &      38,095,415 \\
     & & & &   5 & . & . & . & . &      47,161,435 \\
     & & & &   6 & . & . & . & . &      56,051,398 \\
     & & & &   . & . & . & . & . &     .   .   .   \\
     & & & &  98 & . & . & . & . &     431,055,833 \\
     & & & &  99 & . & . & . & . &     432,490,825 \\
     & & & & 100 & . & . & . & . &     433,897,951 \\
     & & & & 101 & . & . & . & . &     435,277,751 \\
     & & & &   . & . & . & . & . &     .   .   .   \\
     & & & & 118 & . & . & . & . &     455,030,042 \\
     & & & & 119 & . & . & . & . &     455,999,472 \\
     & & & & 120 & . & . & . & . &     456,950,076 \\
     & & & & 121 & . & . & . & . &     457,882,220 \\
     & & & &   . & . & . & . & . &     .   .   .   \\
     & & & & 138 & . & . & . & . &     471,226,168 \\
     & & & & 139 & . & . & . & . &     471,881,080 \\
     & & & & 140 & . & . & . & . &     472,523,275 \\
     & & & & 141 & . & . & . & . &     473,152,998 \\
     & & & &   . & . & . & . & . &     .   .   .   \\
     & & & & 152 & . & . & . & . &     479,322,884 \\
     & & & & 153 & . & . & . & . &     479,820,563 \\
     & & & &   . & . & . & . & . &     .   .   .   \\
     & & & & 199 & . & . & . & . &     494,754,836 \\
     & & & & 200 & . & . & . & . &     494,952,836 \\
  \end{supertabular}
\end{center}

\noindent
['The above table having been calculated very accurately by us the 
undersigned Bookkeepers to My Lords the States-General, each separately, 
and having been collated by us, we find that a perfect agreement exists, 
without there being any error in the figures.

\qquad\qquad\qquad(Signed)

\begin{flushright}
  `T. BELLECHIERE - JACOB LENSE.'\footnote{The above table, computed to 
such a nicety by De Witt's directions, is composed of the progressive
summations of the present values of 1 Million Florins or 20 Million 
Stuyvers per annum, receivable in 100 half-yearly instalments for 50 years.  
The second and every even term will be found correct, on the supposition
of discount at 4 per cent.\ per annum; but the first and every odd term 
erroneous, in the same way that the remark is applicable to Smart's and Tetens'
(or Von Drateln's) Tables, at intermediate half-years, by reason of the
interest being reckoned by a geometric instead of by an arithmetic mean.
In the original a complete table is given from 1 to 200 half-years, which,
however, it is useless to repeat in full, as the even terms may be obtained
by an easy process from the data in other works, and the odd terms are
inapplicable to modern purposes.

\textit{Struyck}, in his \textit{Uitrekening van de Lyfrenten} has some 
remarks on the ``prodigous labor'' of the two bookkeepers who calculated the
Table, although when we compare it with similar ordinary computations of
more modern times it is relatively not worthy of such an appellation.  At
the present date, the tendency is certainly to underestimate such labors;
a reaction to the \textit{juste milieu} may, however, take place after a
surfeit of Statistics.}

\end{flushright}

\medskip

``Thus, then, since an annuitant, having purchased and sunk a life annuity 
upon a young nominee, has in possession, or in his favor, as many 
different expectations or chances as there are half-years in which the 
death of the nominee may occur;--- since the first 100 different expectations 
or chances (comprising the term of 50 years, reckoning from the day of the 
constitution or purchase of the annuity,) may result with the same facility, 
and relatively to their probability are equal;--- since during this term each 
second half-year of the aforesaid nominee's life is equally destructive or 
mortal; (which is demonstrated in the third proposition;) since the 
following 20 chances or expectations (comprising the first 10 years after 
the expiration of the 50 years above cited ), considered one with the 
other, each in proportion to each of the first 100 chances, are not in 
a lower ratio than 2 to 3; (according to the third proposition;)---since 
the 20 expectations or chances of the 10 years after the expectation of 
the first 50 years), also considered one with the other, each in proportion 
to each of the first 100 expectations or chances, are not in a lower ratio 
than 1 to 2; (according to the third presupposition;)---since the 14 
following expectations or chances (comprising the 7 years after the 
expiration of the two preceding decennial terms, the epoch at which we 
here suppose the man to terminate his life), taken one with the other, 
each in proportion to each of the first 100 expectations or chances, are 
not in a lower ratio than 1 to 3;--- it follows that the aforesaid annuitant 
has in possession, or in his favor, more chances or expectations than there 
are in the following table:--

\bigskip

{\small
\begin{center}
  \begin{supertabular}{ccrcc}
    \multicolumn{1}{c}{Chances.}
      &  & of Stuyvers & & The Life to \\
      &  &             & & survive Half-years. \\
         1 & . &           0 & . &  0 \\
         1 & . &   9,805,807 & . &  1 \\
         1 & . &  19,421,192 & . &  2 \\
         1 & . &  28,849,853 & . &  3 \\
         1 & . &  38,095,415 & . &  4 \\
         1 & . &  47,161,435 & . &  5 \\
         1 & . &  56,051,398 & . &  6 \\
           &   &             &   &  $\left.\begin{tabular}{r} 
                                             \text{7 to 97} \\
                                             \text{given in original}
                                             \end{tabular}\right\}$
                                              \\
         1 & . & 431,055,832 & . & 98 \\
         1 & . & 432,490,825 & . & 99 \\
             \cline{3-3}
           & \multicolumn{1}{r}{Sum} & 28,051,475,578
           & \multicolumn{1}{r}{\ \hspace{0.7cm}$\text{Once}=$} 
           & 28,051,475,578\footnote{The addition here presents a clerical 
         error.  It should give 281, \&c., instead of, as above, 280, \&c.;
         and the general summation 409, \&c., instead of 408, \&c.  This
         proceeds no further; for in the working of the annuity valuation,
         the total is correctly given by De Witt.}
         \\ \cline{3-3}
\\ \ \\
      $\frac{2}{3}$ & . & 433,897,951 & . &  100 \\
      $\frac{2}{3}$ & . & 435,277,751 & . &  101 \\
                    &   &             &   &  $\left.\begin{tabular}{r} 
                                                      \text{102 to 117} \\
                                                      \text{in original}
                                                      \end{tabular}\right\}$
                                                        \\
      $\frac{2}{3}$ & . & 455,030,042 & . &  118 \\
      $\frac{2}{3}$ & . & 455,999,472 & . &  119 \\
             \cline{3-3}
           & \multicolumn{1}{r}{Sum} & 8,911,946,713 
           & \multicolumn{1}{r}{\ \hspace{0.7cm}$\text{Two-thirds}=$} 
           & 5,941,297,809 \\ \cline{3-3}
           &   &
           & \multicolumn{1}{r}{\ \hspace{0.7cm}$\text{Carried forward}=$} 
           & 33,992,773,387
\\ \ \\
    \multicolumn{1}{c}{\hspace{-0.4cm}Chances.}
      & & of Stuyvers & & The Life to \\
      & &             & & survive Half-years. \\
      $\frac{1}{2}$ & . & 456,950,076 & . &  120 \\
      $\frac{1}{2}$ & . & 457,882,220 & . &  121 \\
                    &   &             &   &  $\left.\begin{tabular}{r} 
                                                      \text{122 to 137} \\
                                                      \text{in original}
                                                      \end{tabular}\right\}$
                                                        \\
      $\frac{1}{2}$ & . & 471,226,168 & . &  138 \\
      $\frac{1}{2}$ & . & 471,881,080 & . &  139 \\
             \cline{3-3}
           & \multicolumn{1}{r}{Sum} & 9,297,075,282 
           & \multicolumn{1}{r}{\ \hspace{0.7cm}$\text{One-half}=$} 
           & 4,648,537,641 \\ \cline{3-3}
\\ \ \\
      $\frac{1}{3}$ & . & 472,523,275 & . &  \qquad140\qquad \\
      $\frac{1}{3}$ & . & 473,152,998 & . &  \qquad141\qquad \\
                    &   &             &   &  \qquad
                                                    $\left.\begin{tabular}{r} 
                                                      \text{142 to 151} \\
                                                      \text{in original}
                                                      \end{tabular}\right\}$
                                                      \qquad  \\
      $\frac{1}{3}$ & . & 479,322,884 & . &  \qquad152\qquad \\
      $\frac{1}{3}$ & . & 479,820,563 & . &  \qquad153\qquad \\
             \cline{3-3} \\
           & \multicolumn{1}{r}{Sum} & 6,668,408,125 
           & \multicolumn{1}{r}{\ \hspace{0.7cm}$\text{One-third}=$} 
           & \qquad2,222,802,708\qquad \\ \cline{5-5}
           &   &
           & \multicolumn{1}{l}{Total} & \qquad40,864,113,736\qquad \\ 
             \cline{5-5}
  \end{supertabular}
\end{center}
}

\begin{center}
  \footnote{40,964,113,736 is here correctly given by De Witt.}
  40,964,113,736 divided by 128 gives 320,032,130 8 9-16, which \\
  divided by 20 gives 16,001,606 18-9.
\end{center}

``Whence it follows that we can immediately determine, by a mathematical 
calculation, according to the principle of the second proposition above 
enunciated, the worth to the aforesaid annuitant of all the above-mentioned 
chances, taken together, always presupposing that such value is payable 
in ready money on the day of purchase of the annuity; and the method 
is as follows:--

``Since the first 100 items, each taken once, or each multiplied by the 
number 1, form the sum of 28,151,475,578 stuyvers; since the 20 following 
items, two-thirds of each being taken, or each multiplied by 2/3 
(or, which is the same thing, two-thirds of the sum of the aforesaid 
20 items,) produce a sum of 5,941,297,809 stuyvers; since then the half 
of the 20 following items gives a sum of 4,648,537,641 stuyvers, and the 
third of the 14 following and last items that of 2,222,802,708 stuyvers; 
these sums being combined, amount together to the sum of 40,964,113,736 
stuyvers; which being divided by 128 (that is to say, the real and exact 
value of all the collective chances,) the sum of 320,032,132 stuyvers, 
or 16,001,607 florins: so that 1,000,000 per annum of life annuities, 
sunk or purchased on a young life should consequently be sold for more 
than 16,001,607 florins,\footnote{De Witt's calculation may be simplified and 
explained as follows: \textit{Firstly.}  Out of 128 lives, aged say 3 years, 
1 is supposed to die in every half-year of the first hundred half-years,
or 2 per annum for 50 years, leaving 28 alive, aged 53 years, at the end of
the term; out of whom 1 dies in every 9 months, being 0.6\.6 per half-year 
during the next 20 half years, or 1.3\.3 per annum for 10 years, leaving 
16.6\.6 alive aged 63 years at the end of the second term; of whom 1 dies in
every year for 10 years, being 0.5 per half-year during the next 20 half-years,
leaving 5.6\.6 alive aged 73 at the end of the third term; of whom 1 dies
in every year-and-a-half for 7 years, being 0.3\.3 per half-year during
the next fourteen years, leaving 1 alive aged 80 at the end of the fourth 
term; which survivor does not live over another half-year.  \textit{Secondly.}
Out of the 128 lives, those who die in the respective half-years between
the ages of 3 and 80, will receive an annuity certain in half-yearly
instalments. for a term equal to the number of \textit{completed} half-years
elapsed between age 3 and the date of their death; therefore, the sum of the
present values of half-yearly annuities certain, for the corresponding
terms multiplied into the numbers \textit{dying} within such respective
terms, gives the present worth of all the annuities which will be enjoyed
by the 128 lives, one-hundredth and twenty-eighth of which represents the 
present value of the single-life annuity at age of, say, 3 years.  The
system of valuation is therefore identical with the fifth method described 
by \textit{Tetens}, whose formula I have had the pleasure to refer to on a
previous occasion.  (See the \textit{Assurance Magazine}, No.\ 1, pp.\ 9 and 
18; and No.\ 11, p.\ 18).

If arranged in the modern form of a life table, the following abstract would 
represent the course of the results of De Witt's suppositions as to
mortality.

\begin{center}
  \begin{tabular}{rlll}
  \multicolumn{1}{c}{Half-year Number.} &
  \multicolumn{1}{c}{Age.} &
  \multicolumn{1}{c}{Number of living.} &
  \multicolumn{1}{c}{Decrements.}            \\
     1 &   3             & 128      & 1      \\
     2 & $3\frac{1}{2}$  & 127      & 1      \\
     . &   .             &   .      & .      \\
    99 & $52\frac{1}{2}$ &  29      & 1      \\
   100 &  53             &  28      & 0.6\.6 \\
   101 & $53\frac{1}{2}$ &  27.3\.3 & .      \\
     . &  .              &  26.6\.6 & 0.6\.6 \\
   120 &  63             &  15.6\.6 & 0.50   \\
   121 & $63\frac{1}{2}$ &  15.1\.6 & 0.50   \\
     . &   .             &   .      & .      \\
   140 &  73             &   5.6\.6 & 0.33   \\
     . &   .             &   .      & .      \\
   154 &  80             &   1.00   & 1.00
  \end{tabular}
\end{center}} preserving the right proportion above mentioned, 
i.e., that proportionately each florin per annum of life annuity is worth 
more at 16 florins than the interest of a redeemable annuity at 4 per 
cent.\ per annum,- and consequently the person who for 16 florins has 
purchased a young, vigorous, and healthy life, has made a remarkably 
advantageous contract; I assert it to be remarkably advantageous for the 
following reasons:--

``Because, in the first place, we have not been able to rate at a certain 
price, by perfect calculation or correct estimation, the power which the 
annuitant possesses (power which is of very great value to him) of choosing 
a life, or person in full health, and with a manifest likelihood of 
prolonged existence, upon whom to constitute or purchase his annuity, 
and there is much less risk or danger of a select, vigorous, and healthy 
life, dying in the first half-year than in some of the following half-years 
at the beginning of which the aforesaid life might perhaps prove to be 
in a weak state of health or even in a fatal illness; and such greater 
likelihood of prolongation of life in the the purchase of an annuity upon 
a select, healthy, and robust life, may further extend itself to the the 
second, third, and some following terms or half-years.

``In the second place, the advantage resulting from the aforesaid selection 
is so much more considerable, as one half year of life, at the commencement 
of and shortly after the purchase of the life annuity, is of greater value 
to the annuitant, with respect to the price of the purchase, than eighteen 
half-years during which the person upon whom the annuity is purchased might 
live after the said purchase, from the age, for example, of 70 to 79 
years,---a circumstance which, although at first sight it might appear 
strange and paradoxical, is nevertheless real and susceptible of 
demonstration.

``In the third place, although each of the first 100 half years expiring 
after the purchase be considered as equally destructive or mortal, 
according to the principle of the before-established calculation, by 
reason of the scarcely appreciable difference existing between the first 
and second half of each year, it is, however, certain, when we examine 
the matter very scrupulously, that the likelihood of decrease of the 
nominees upon whom life annuities are usually purchased is less purchase 
than in the subsequent years, seeing that the said life annuities are 
oftenest purchased and sunk upon the lives of young and healthy children 
of 3, 4, 5, 6, 7, 8, 9, 10 years, or thereabouts.  During that time, and 
for some years ensuing, these young lives, having become more robust, are 
less subject to mortality than about 50 years afterwards, and than for 
some years anterior to these 50 years; and so much the more, as during the 
first aforesaid years they either are not, or are but little, exposed to 
external accidents and extraordinary causes of death, such as those from 
war, dangerous voyages, debauch, or excess of drink, of the sex, and other 
dangers;--- for females, there are also confinements and other like 
causes;--- so that the first years after the purchase or foundation of the 
annuity are the least dangerous, which is a considerable advantage for the 
annuitant, particularly if we reflect, as I have above stated, that one 
of the said first years may, as regards the original price of purchase, 
balance a great number of subsequent years.

``Finally, and in the fourth place, it might also evidently occur, that 
the life upon which the annuity has been sunk were to live more than 
77 years after the purchase, being the time supposed in the above 
calculation as the term of human life, although such considerations 
cannot be of much importance; for, notwithstanding that by presupposing 
the aforesaid nominee living still longer than the expiration of the said 
term, and preserving life up to the hundredth year inclusive, so that the 
annuitant or his heirs were to receive 46 more entire half years of annuity, 
after the expiration of the aforesaid 77 years, this could not, however, 
increase the price of the life annuity (calculated, as precedes, at about 
16 years' purchase, i.e. at more than 16 florins of capital for 1 florin 
of annuity per annum,) by more than 14 stuyvers of the same capital; and 
even if the annuitant were, after the expiration of the above 100 years, 
to enjoy the life annuity from half year to half year, and that perpetually, 
the value of the capital at the time of first purchase would not thereby be 
increased by 10 stuyvers.

``Whence likewise, although it may be considered that the latter years are 
not established as sufficiently destructive and mortal in the aforesaid 
presuppositions and in the calculations upon which I have based them, when 
compared with the anterior years and the time of life's vigor, we easily 
conclude that it could not cause an appreciable rise in the price of the 
purchase found by the above calculation, which in fact is true, even on 
the presupposition of each half year of the 10 years after the sixtieth 
year of purchase being, instead of twice, three times more destructive 
and mortal than each half year of the first 50 years, and of each half 
year of the 7 subsequent years being, instead of three times, five times 
more destructive and mortal than each of the aforesaid first years; and 
even on the presupposition again, as above, that the said nominee would 
not survive beyond 77 years after the first purchase.  All these 
presuppositions (which, however, manifestly represent the life as subject 
to too high mortality) could scarcely reduce by 6 stuyvers the aforesaid 
16 florins or value of the before-described annuity.  In consequence of 
all these reasons, we may assume it as established and demonstrated, 
that the value of a life annuity, in proportion to the redeemable annuity 
at 25 years' purchase, is really not below, but certainly above 16 years' 
purchase; so that a person, wishing to purchase a life annuity in such 
proportion and according to its real value, ought to pay more than 16 
florins for 1 florin of annuity per annum.

``Besides the consideration that this calculation has been made on the 
principle of a deduction of 4 per cent.\ per annum, at compound interest, 
and this with such benefit to the purchaser of the life annuity that he 
would realize not only the interest per annum, but also, without any 
intermission, interest upon interest at 4 per cent.\ per annum, as though 
he could always thus advantageously make use of his money in purchase of 
annuity; it is constant that one could not always find such opportunity 
of investing it, and that one is sometimes obliged to let it lie fallow 
for some time, and often to lend it at a materially smaller interest, 
to provide against a greater loss.

``Even besides this, as the capital of life annuities is not subject to 
taxation, not to a reduction to a lower amount of annuity or interest, 
it follows, that if the blessing of the Almighty continue to be vouchsafed 
to this country, we may consider the life annuity as much more advantageous 
to the annuitant than the redeemable annuity, as may manifestly be judged 
by the example of foregoing times,-- by reflecting, in fact that My Lords 
the States of Holland and West Friesland have in the course of a few years 
not only increased the charge for life annuities from 11 years' purchase 
to 12 years' purchase, and from 12 years' purchase to 14 years' purchase, 
but that these annuities have been sold, even in the present century, first 
at 6 years' purchase, then at 7 and at 8, and that the majority of all 
life annuities now current and at the country's expense were obtained at 
9 years' purchase; which annuities, by reason of the successive reductions 
of the rate of interest from 6 to 5 per cent., and then from 5 per 
cent.\ to 4 per cent., produce to the annuitants an actual profit of nearly 
one-half of each half-year's payment, and of more than one-half in the case 
of those annuities which were obtained at 8 years' purchase or under.

\begin{flushright}
  ``JOHN DE WITT.''
\end{flushright}

\end{document}

%