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{\Large \textbf{\textit{An Estimate of the Degrees of the Mortality of Mankind, 
drawn from curious} Tables \textit{of the} Births \textit{and} Funerals
\textit{at the City of} Breslaw ; \textit{with an Attempt to ascertain
the Price of} Annuities \textit{upon} Lives. \textit{By Mr.}\ E.\ 
Halley, \textit{R.S.S.}}}

\bigskip

The  Contemplation of the \textit{Mortality} of \textit{Mankind}, has
besides the \textit{Moral}, its \textit{Physical} and \textit{Political}
Uses, both which have been some years since most judiciously considered
by the curious \textit{Sir William Petty}, in his \textit{Natural} and
\textit{Political} Observations on the Bills of \textit{Mortality} of
\textit{London}, owned by Captain \textit{John Graunt}. And since in a
like Treatise on the Bills of \textit{Mortality} of [596] \textit{Dublin}. But
the Deduction from those Bills of \textit{Mortality} seemed even to
their Authors to he defective: First, In that the \textit{Number} of the
People was wanting.  Secondly, That the \textit{Ages} of the People 
dying was not to be had. And Lastly, That both \textit{London} and
\textit{Dublin} by reason of the great and casual Accession of
\textit{Strangers} who die therein, (as appeared in both, by the great
Excess of the \textit{Funerals} above the \textit{Births}) rendered them
incapable of being Standards for this purpose; which requires, if it
were possible, that the People we treat of should not at all be changed,
but die where they were born, without any Adventitious Increase from
Abroad, or Decay by Migration elsewhere. 

This \textit{Defect} seems in a great measure to be satisfied by the late
curious Tables of the Bills of \textit{Mortality} at the City of
\textit{Breslaw}, lately communicated to this honourable Society by
Mr.\ \textit{Justell}, wherein both the \textit{Ages} and \textit{Sexes}
of all that die are monthly delivered, and compared with the number of
the Births, for Five Years last past, viz.\ 1687, 88, 89, 90, 91, 
seeming to be done with all the Exactness and Sincerity possible. 

This City of \textit{Breslaw} is the Capital City of the Province of
\textit{Silesia}; or, as the Germans call it, \textit{Schlesia}, and is
situated on the Western Bank of the River \textit{Oder}, anciently
called \textit{Viadrus}; near the Confines of \textit{Germany} and
\textit{Poland}, and very nigh the Latitude of \textit{London}. It is
very far from the Sea, and as much a \textit{Mediterranean} Place as can
be desired, whence the Confluence of Strangers is but small, and the
Manufacture of Linnen employs chiefly the poor People of the place, as
well its of the Country round about; whence comes that sort of Linnen we
usually call your \textit{Sclesie Linnen}; which is the chief, if not
the only Merchandize of the place. For these Reasons the People of this
City seem most pro-[597]per for a Standard; and the rather, for that the
Births do, a small matter, exceed the \textit{Funerals}. The only thing
wanting is the Number of the whole People, which in some measure I have
endeavoured to supply by comparison of the Mortality of the People of
all Ages, which I shall from the said Bills trace out with all the
Accuracy possible. 

It appears that in the Five Years mentioned, viz.\ from 87 to 91
inclusive, there were \textit{born} 6193 Persons, and buried 5869; that
is, born \textit{per Annum} 1238, and \textit{buried} 1174; whence an
\textit{Encrease} of the People may be argued of 64 per Annum, or of about a
20th part, which may perhaps be ballanced by the Levies for the
\textit{Emperor}'s Service in his Wars.  But this being contingent, and
the Births certain, I will suppose the People of \textit{Breslaw} to be
encreased by 1238 Births annually.  Of these it appears by the same
Tables, that 348 do die \textit{yearly} in the \textit{first Year} of
their \textit{Age}, and that but 890 do arrive at a full \textit{Years
Age}; and likewise, that 198 do die in the Five Years between 1 and 6
compleat, taken at a \textit{Medium}; so that but 692 of the Persons
born do survive \textit{Six} whole \textit{Years}.  From this
\textit{Age} the Infants being arrived at some degree of Firmness,
grow less and less \textit{Mortal}; and it appears that of the whole
People of \textit{Breslaw} there die \textit{yearly}, as in the
following Table, wherein the upper Line shews the \textit{Age}, and the
next under it the \textit{Number} of Persons of that Age \textit{dying
yearly}.[598] 
\[ \begin{array}{c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c}
\phantom{0}7&amp;.&amp;\phantom{0}8&amp;9&amp;.&amp;&amp;.&amp;&amp;14&amp;&amp;.&amp;18&amp;.&amp;21&amp;.&amp;27&amp;.&amp;28&amp;.&amp;.&amp;&amp;35&amp;. \\
11&amp;.&amp;11&amp;.&amp;6&amp;.&amp;5\frac{1}{2}&amp;.&amp;\phantom{0}2&amp;.&amp;3\frac{1}{2}&amp;\phantom{0}5&amp;6&amp;
4\frac{1}{2}&amp;6\frac{1}{2}&amp;\phantom{0}9&amp;.&amp;\phantom{0}8&amp;.&amp;7&amp;.&amp;\phantom{0}7&amp;.
   \end{array} \]
\[ \begin{array}{c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c}
36 &amp; .&amp;&amp;42&amp;.&amp;&amp;&amp;45&amp;&amp;&amp;\ &amp;&amp;49&amp;54&amp;.&amp;\ \ &amp;55&amp;.&amp;56&amp;&amp;\phantom{0}.&amp;&amp;63\\
\phantom{0}8&amp;.&amp;9\frac{1}{2}&amp;\phantom{0}8&amp;.&amp;9&amp;.&amp;\phantom{0}7&amp;.&amp;7&amp;\ &amp;.&amp;10
&amp;11&amp;.&amp;\ \ &amp;\phantom{0}9&amp;.&amp;\phantom{0}9&amp;.&amp;10&amp;.&amp;12
   \end{array} \]
\[ \begin{array}{c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }
                 c@{\ \ }c@{\ \ }c}
&amp;70&amp;71&amp;.&amp;72&amp;&amp;\ &amp;77&amp;&amp;&amp;&amp;81&amp;&amp;&amp;\ &amp;84&amp;.&amp;&amp;&amp;90&amp;91&amp;\phantom{0}. \\
9\frac{1}{2}&amp;14&amp;9\phantom{0}&amp;.&amp;11&amp;9\frac{1}{2}&amp;&amp;6&amp;.&amp;7&amp;.&amp;3&amp;.&amp;4&amp;.
 &amp;\phantom{0}2&amp;.&amp;1&amp;.&amp;1\,.&amp;1\phantom{0}&amp;.
   \end{array} \]
\begin{flushleft}\qquad
\begin{tabular}{c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c}
\ \ \ \ \ \ &amp;98&amp;.&amp;99&amp;.&amp;100.\\
&amp;\phantom{0}0&amp;.&amp;$\frac{1}{5}$&amp;.&amp;$\frac{3}{5}$
   \end{tabular}\end{flushleft}

And where no \textit{Figure}, is placed over, it is to be understood of
those that die between the Ages of the preceeding and consequent
\textit{Column}.

From this Table it is evident, that from the Age of 9 to about 25 there
does not die above 6 \textit{per Annum} of each \textit{Age}, Which is
much about one \textit{per Cent.}\ of those that are of those
\textit{Ages}: And whereas in the 14, 15, 6, 17 \textit{Years} there
appear to die much fewer, as 2 and $3\frac{1}{2}$, yet that seems rather
to be attributed to Chance, as are the other Irregularities in the
Series of Ages, which would rectifie themselves, were the number of
Years much more considerable, as 20 instead of 5.  And by our own
Experience in \textit{Christ-Church Hospital}, I am informed there die
of the \textit{Young Lads}, much about one \textit{per Cent.\ per
Annum}, they being of the foresaid Ages.  From 25 to 50 there seem to
die from 7 to 8 and 9 per Annum of each Age; and after that to 70, they
growing more \textit{crasie}, though the \textit{Mortality encreases},
and there are found to die 10 or 11 of each Age \textit{per Annum}: From
thence the number of the \textit{Living} being grown very  small they [599]
gradually decline till there be none left to  \textit{die}; as may be
seen at one View in the Table. 

From these Considerations I have formed the \textit{adjoyned Table},
whose Uses are manifold, and give a more just \textit{Idea} of the
\textit{State} and \textit{Condition} of \textit{Mankind}, than any
thing yet extant that I know of.  It exhibits the \textit{Number} of
\textit{People} in the City of \textit{Breslaw} of all Ages, from the
Birth to extream \textit{Old Age}, and thereby shews the Chances of
\textit{Mortality} at all \textit{Ages}, and likewise how  to make a
certain Estimate of the value of \textit{Annuities} for \textit{Lives},
which hitherto has been only done by an imaginary \textit{Valuation}. 
Also the \textit{Chances} that there are that a \textit{Person} of any
\textit{Age} proposed does live to any other Age given; with many more,
as I shall hereafter shew.  This \textit{Table} does shew the
\textit{number} of \textit{Persons} that are living in the \textit{Age}
current annexed thereto, as follows: 
\begin{center}
{\footnotesize
  \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
    \hline
Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- \\
Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.\\
\hline
 \z1 &amp;1000 &amp; \z8 &amp; 680 &amp;  15 &amp; 628 &amp;  22 &amp; 586 &amp;  29 &amp; 539 &amp;  36 &amp; 481 \\
 \z2 &amp;\z855&amp; \z9 &amp; 670 &amp;  16 &amp; 622 &amp;  23 &amp; 579 &amp;  30 &amp; 531 &amp;  37 &amp; 472 \\
 \z3 &amp;\z798&amp;  10 &amp; 661 &amp;  17 &amp; 616 &amp;  24 &amp; 573 &amp;  31 &amp; 523 &amp;  38 &amp; 463 \\
 \z4 &amp;\z760&amp;  11 &amp; 653 &amp;  18 &amp; 610 &amp;  25 &amp; 567 &amp;  32 &amp; 515 &amp;  39 &amp; 454 \\
 \z5 &amp;\z732&amp;  12 &amp; 646 &amp;  19 &amp; 604 &amp;  26 &amp; 560 &amp;  33 &amp; 507 &amp;  40 &amp; 445 \\
 \z6 &amp;\z710&amp;  13 &amp; 640 &amp;  20 &amp; 598 &amp;  27 &amp; 553 &amp;  34 &amp; 499 &amp;  41 &amp; 436 \\
 \z7 &amp;\z692&amp;  14 &amp; 634 &amp;  21 &amp; 592 &amp;  28 &amp; 546 &amp;  35 &amp; 490 &amp;  42 &amp; 427 \\
\hline
Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- &amp;Age. &amp;Per- \\
Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.&amp;Curt.&amp;sons.\\
\hline
  43 &amp; 417 &amp;  50 &amp; 346 &amp;  57 &amp; 272 &amp;  64 &amp; 202 &amp;  71 &amp; 131 &amp;  78 &amp;\z58 \\
  44 &amp; 407 &amp;  51 &amp; 335 &amp;  58 &amp; 262 &amp;  65 &amp; 192 &amp;  72 &amp; 120 &amp;  79 &amp;\z40 \\
  45 &amp; 397 &amp;  52 &amp; 324 &amp;  59 &amp; 252 &amp;  66 &amp; 182 &amp;  73 &amp; 109 &amp;  80 &amp;\z41 \\
  46 &amp; 387 &amp;  53 &amp; 313 &amp;  60 &amp; 242 &amp;  67 &amp; 172 &amp;  74 &amp;\z98 &amp;  81 &amp;\z34 \\
  47 &amp; 377 &amp;  54 &amp; 302 &amp;  61 &amp; 232 &amp;  68 &amp; 162 &amp;  75 &amp;\z88 &amp;  82 &amp;\z28 \\
  48 &amp; 367 &amp;  55 &amp; 292 &amp;  62 &amp; 222 &amp;  69 &amp; 152 &amp;  76 &amp;\z78 &amp;  83 &amp;\z23 \\
  49 &amp; 357 &amp;  56 &amp; 282 &amp;  63 &amp; 212 &amp;  70 &amp; 142 &amp;  77 &amp;\z68 &amp;  84 &amp;\z20 \\
\hline
  \end{tabular}
}

  \begin{longtable}{r@{\qquad\qquad}r}
    Age.&amp;Persons.\\
      7 &amp;  5547 \\
     14 &amp;  4584 \\
     21 &amp;  4270 \\
     28 &amp;  3964 \\
     35 &amp;  3604 \\
     42 &amp;  3178 \\
     49 &amp;  2709 \\
     56 &amp;  2194 \\
     63 &amp;  1694 \\
     70 &amp;  1204 \\
     77 &amp;   692 \\
     84 &amp;   253 \\
    100 &amp;   107 \\
    \hline
        &amp; 34000 \\
    \hline
    \multicolumn{2}{r}{Sum Total.} \\
  \end{longtable}
\end{center}

Thus it appears, that the whole People of \textit{Breslaw} does consist
of 34000 \textit{Souls}, being the Sum \textit{Total} of the Persons of
all Ages in the \textit{Table}: The first use hereof [600] is to shew the
Proportion of \textit{Men} able to bear \textit{Arms} in any
\textit{Multitude}, which are those between 18 and 56, rather than 16
and 60; the one being generally too weak to bear the \textit{Fatigues}
of \textit{War} and the Weight of \textit{Arms}, and the other too
crasie and infirm from \textit{Age}, notwithstanding particular
Instances to the contrary. Under 18 from the \textit{Table}, are found
in this City 11997 Persons, and 3950 above 56, which together make
15947.  So that the Residue to 84000 being 18053 are Persons between
those \textit{Ages}.  At least one half thereof are Males, or 9027.  So
that the whole Force this City can raise of \textit{Fencible Men}, as
the Scotch call them, is about 9000 or 9/34, or somewhat more than a
quarter of the \textit{Number} of \textit{Souls}, which may perhaps pass
for a Rule for all other places. 

The \textit{Second Use} of this Table is to shew the differing degrees
of \textit{Mortality}, or rather \textit{Vitality} in all \textit{Ages};
for if the number of Persons of any \textit{Age}, remaining after one
year, be divided by the difference between that and the number of the
\textit{Age} proposed, it shows the \textit{odds} that there is, that a
Person of that Age does not die in a \textit{Year}.  As for Instance, a
Person of 25 \textit{Years} of \textit{Age} has the odds of 560 to 7 or
80 to 1, that he does not \textit{die}, in a Year: Because that of 567,
living of 25 years of Age, there do die no more than 7 in a
\textit{Year}, leaving 560 of 26 Years old. 

So likewise for the \textit{odds}, that any Person does not die before
lie attain any proposed \textit{Age:} Take the \textit{number} of the
remaining Persons of the Age proposed, and divide it by the difference
between it and the number of those of the Age of the Party proposed; and
that shews the \textit{odds} there is between the Chances of the Party's
living or dying. As for Instance; What is the \textit{odds} that a Man
of 40 lives 7 Years: Take the number of Persons of 47 years, which in
the Table is 377, and [601] substract it from the number of Persons of 40
years, which is 446, and the \textit{difference} is 68: Which shews that
the \textit{Persons dying} in that 7 years are 68, and that it is 377 to
68 or 51 to 1, that a Man of 40 does live 7 Years.  And the like for any
other \textit{number} of \textit{Years}. 

\textit{Use} III. But if it be enquired at what number of
\textit{Years}, it is an even Lay that a Person of any \textit{Age}
shall die, this Table readily performs it.  For if the \textit{number}
of Persons living of the \textit{Age} proposed be \textit{halfed}, it
will be found by the \textit{Table} at what Year the said
\textit{number} is reduced to half by \textit{Mortality}; and that is
the Age, to which it is an even Wager that a Person of the \textit{Age}
proposed shall arrive before he \textit{die}. As for Instance; A Person
Of 30 Years of Age is proposed, the number of that Age is 531, the half
thereof is 265, which number I find to be between 57 and 58 Years; so
that a Man of 30 may reasonably expect to live between 27 and 28 Years. 

\textit{Use} IV. By what has been said, the \textit{Price} of
\textit{Insurance} upon \textit{Lives} ought to be regulated, and the
difference is discovered between the \textit{price} of ensuring the
\textit{Life} of a Man of 20 and 50, for Example: it being 100 to 1
that a Man of 20 dies not in a year, and but 38 to 1 for a Man of 50
Years of Age. 

\textit{Use} V.  On this depends the Valuation of \textit{Annuities}
upon \textit{Lives}; for it is plain that the purchaser ought to pay for
only such a part of the value of the \textit{Annuity}, as he has Chances
that he is living; and this ought to be computed Yearly, and the Sum of
all those yearly Values being added together, will amount to the value
of the \textit{Annuity} for the \textit{Life} of the Person proposed.
Now the present value of money payable after a term of years, at any
given rate of Interest, either may be had from Tables already computed;
or almost as compendiously, [602] by the Table of Logarithms: For the
Arithmetical Complement of the Logarithm of Unity and its yearly
Interest (that is, of 1,\,06 for Six \textit{per Cent.}\ being 
9,\,974694.) being multiplied by the number of years proposed, gives the
present value of One Pound payable after the end of so many years.  Then
by the foregoing Proposition, it will be as the number of Persons living
after that term of years, to the number dead; so are the Odds that any
one Person is Alive or Dead.  And by consequence, as the Sum of both or
the number of Persons living of the Age first proposed, to the number
remaining after so many years (both given by the Table) so the present
value of the yearly Sum payable after the term proposed, to the Sum
which ought to be paid for the Chance the person has to enjoy the
Annuity after so many years.  And this being repeated for every year of
the persons Life, the Sum of all the present Values of those Chances is
the true Value of the Annuity.  This will without doubt appear to be a 
most laborious Calculation, but it being one of the principal Uses of
this Speculation, and having found some \textit{Compendia} for the Work,
I took pains to compute the following Table, being the short Result of a
not ordinary number of Arithmetical Operations; It shows the Value of
Annuities for every Fifth Year of Age, to the Seventieth, as follows.
\begin{center}
  \begin{tabular}{|c|c|c|c|c|c|}
    \hline
    Age. &amp; Years Purchase. &amp; Age. &amp; Years Purchase. &amp; Age. &amp; Years Purchase. \\
    \hline
     \z1 &amp;    10,28        &amp;  25  &amp;    12,27        &amp;  50  &amp;     9,21 \\       
     \z5 &amp;    13,40        &amp;  30  &amp;    11,72        &amp;  55  &amp;     8,51 \\
     10  &amp;    13,44        &amp;  35  &amp;    11,12        &amp;  60  &amp;     7,60 \\
     15  &amp;    13,33        &amp;  40  &amp;    10,57        &amp;  65  &amp;     6,54 \\
     20  &amp;    12,78        &amp;  45  &amp;   \z9,91        &amp;  70  &amp;     5,32 \\
    \hline
  \end{tabular}
\end{center}
\begin{flushright}
  [603]
\end{flushright}

This shews the great Advantage of putting Money into the present
\textit{Fund} lately granted to their Majesties, giving 14 \textit{per
Cent.\ per Annum}, or at the rate of 7 years purchase for a Life; when
young Lives, at the usual rate of Interest, are worth about 13 years
Purchase.  It shews likewise the Advantage of young Lives over those in
Years; a Life of Ten Years being worth almost $13\frac{1}{2}$ years
purchase, whereas one of 36 is worth but 11.

\textit{Use} V. Two Lives are likewise valuable by the same Rule; for
the number of Chances of each single Life, found in the Table, being
multiplied together, become the Chances of the Two Lives.  And after any
certain Term of Years, the Product of the two remaining Sums is the
Chances that both the Persons are living.  The Product of the two
Differences, being the numbers of the Dead of both Ages, are the Chances
that both the Persons are dead.  And the two Products of the remaining
Sums of the one Age multiplied by those dead of the other, shew the
Chances that there are that each Party survives the other: Whence is
derived the Rule to estimate the value of the Remainder of one Life
after another.  Now as the Product of the Two Numbers in the Table for
the Two Ages proposed, is to the difference between that Product and the
Product of the two numbers of Persons deceased in any space of time, so
is the value of a Sum of Money to be paid after so much time, to the
value thereof under the Contingency of Mortality.  And as the aforesaid
Product of the two Numbers answering to the Ages proposed, to the
Product of the Deceased of one Age multiplied by those remaining alive
of the other; So the Value of a Sum of Money to be paid after any time
proposed, to the value of the Chances that the one Party has that he
survives the other whose number of Deceased you made use of, in the
second Term of the proportion.  This perhaps [604] may be better understood,
by putting $N$ for the number of the younger Age, and $n$ for that of
tile Elder; $Y$, $y$ the deceased of both Ages respectively, and $B$,
$r$ for the Remainders; and $R + Y = N$ and $r + y = n$. Then shall $N$
$n$ be the whole number of Chances; $N$ $n - Y$ $y$ be the Chances that
one of the two Persons is living, $Y$ $y$ the Chances that they are both
dead; $R$ $y$ the Chances that the elder Person is dead and the younger
living; and $r$ $Y$ the Chances that the elder is living and the younger
dead.  Thus two Persons of 18 and 35 are proposed, and after 8 years
these Chances are required.  The Numbers for 18 and 35 are 610 and 490,
and there are 50 of the First Age dead in 8 years, and 73 of the Elder
Age. There are in all $610 \times 490$ or 298900 Chances; of these there
are $50 \times 73$ or 3650 that they are both dead. And as 298900, to
$298900 - 3650$, or 295250: So is the Present value of a Sum of Money to
be paid after 8 years, to the present value of a Sum to be paid if
either of the two live.  And as $560 \times 73$, so are the Chances that
the Elder is dead, leaving the Younger; and as $417 \times 50$, so are
the Chances that the Younger is dead, leaving the Elder.  Wherefore as
$610 \times 490$ to $560 \times 73$, so is the present value of a Sum to
be paid at eight years end, to the Sum to be paid for the Chance of the
Youngers Survivance; and as $610 \times 490$ to $417 \times 50$, so is
the same present value to the Sum to be paid for the Chance of the
Elders Survivance.  This possibly may be yet better explained by
expounding these Products by Rectangular Parallelograms, as in
\textit{Fig}.\ 7.\ wherein $A$ $B$ or $C$ $D$ represents the number of
persons of the younger Age, and $D$ $E$, $B$ $H$ those remaining alive
after a certain term of years; whence $c$ $E$ will answer the number of
those dead in that time: So $A$ $C$, $B$ $D$ may represent the the
number [605] of the Elder Age; $A$ $F$, $B$ $I$ the Survivors after the same
term; and $C$ $F$, $D$ $I$, those of that Age that are dead at that
time.  Then shall the whole Parallelogram $A$ $B$ $C$ $D$ be $N$ $n$ or
the Product of the two Numbers of persons, representing such a number of
Persons of the two Ages given; and by what was said before, after the
Term Proposed the Rectangle $H$ $D$ shall be as the number of Persons of
the younger Age that survive, and the Rectangle $A$ $E$ as the number of
those that die.  So likewise the Rectangles $A$ $I$, $F$ $D$ shall be as
the Numbers, living and dead, of the other Age.  Hence the Rectangle $H$
$I$ shall be as an equal number of both Ages surviving.  The Rectangle
$F$ $E$ being the Product of the deceased, or $Y$ $y$, an equal number
of both dead.  The Rectangle $G$ $D$ or $B$ $y$, a number living of the
younger Age, and dead of the Elder.  And the Rectangle $A$ $G$ or $r$
$Y$ a number living of the Elder Age, but dead of the younger.  This
being understood, it is obvious, that as the whole Rectangle $A$ $D$ or
$N$ $n$ is to the \textit{Gnomon} $F$ $A$ $B$ $D$ $E$ $G$ or $N$ $n - Y$
$y$ so is the whole number of Persons or Chances, to the number of
Chances that one 01 the two persons is living: And as $A$ $D$ or $N$ $n$
is to $F$ $E$ or $Y$ $y$, so are all the Chances, to the chances that
both are dead; whereby may be computed the value of the Reversion after
both Lives. And as $A$ $D$ to $G$ $D$ or $R$ $y$, so the whole number of
Chances, to the Chances that the younger is living and the other dead;
whereby may be cast up what value ought to be paid for the Reversion of
one Life after another, as the case of providing for Clergy-mens widows
and others by such Reversions.  And as $A$ $D$ to $A$ $G$ or $r$ $Y$, so
are all the Chances, to those that the Elder survives the younger. I
have been the more particular, and perhaps tedious, in this matter,
because it is the Key to the Case of Three Lives, which of it self
would not have been so easie to comprehend. 

\begin{figure}
  \begin{center}
    \epsfig{file=halley_fig_1.eps,width=7cm,width=7cm,angle=270,clip=}
  \end{center}
\end{figure}

VII. If Three Lives are proposed, to find the value of an Annuity during
the continuance of any of those three Lives. The Rule is, \textit{As the
Product of the continual multiplication of the Three Numbers, in the
Table, answering to the Ages proposed, is to the difference of that
Product and of the Product of the Three Numbers of the deceased of those
Ages, in any given term of Years; So is the present value of a Sum of
Money to be paid certainly after so many Years, to the present value of
the} [606] \textit{same Sum to be paid, provided one of those three
Persons be, living at the Expiration of that term}. Which proportion
being yearly repeated, the Sum of all those present values will be the
value of an Annuity granted for three such Lives.  But to explain this,
together with all the Cases of Survivance in three Lives: Let $N$ be the
Number in the Table for the Younger Age, $n$ for the Second, and $\nu$
for the Elder Age; let $Y$ be those dead of the Younger Age in the term
proposed, $y$ those dead of the Second Age, and $\upsilon$ those of the
Elder Age; and let $R$ be the Remainder of the younger Age, $r$ that of
the middle Age, and $\varrho$ the Remainder of the Elder Age. Then shall
$R + Y$ be equal to $N$, $r + y$ to $n$, and $\varrho + \upsilon$ to
$\nu$, and the continual Product of the Numbers $N$ $n$ $\nu$ shall be
equal to the continual Product of $R + Y \times r + Y \times \varrho +
\upsilon$\footnote{\textsc{Editor's Note}: This is a misprint for $R + Y
\times r + y \times \varrho + \upsilon$}, which being the whole number
of Chances for three Lives is compounded of the eight Products
following.  (1) $R$ $r$ $\varrho$, which is the number of Chances that
all three of the Persons are living.  (2) $r$ $\varrho$ $Y$, which is
the number of Chances that the two Elder Persons are living, and the
younger dead.  (3) $R$ $\rho$ $y$\footnote{\textsc{Editor's Note} This
is a misprint for $R$ $\varrho$ $y$} the number of Chances that the
middle Age is dead, and the younger and Elder living.  (4) $R$ $r$
$\upsilon$ being the Chances that the two younger are living, and the
elder dead.  (5) $\varrho$ $Y$ $y$ the Chances that the two younger are
dead, and the elder living.  (6) $r$ $Y$ $\upsilon$ the Chances that the
younger and elder are dead, and the middle Age living.  (7) $R$ $y$
$\upsilon$, which are the Chances that the younger is living, and the
two other dead. And Lastly and Eighthly, $Y$ $y$ $\upsilon$, which are
the Chances that all three are dead.  Which latter subtracted from the
whole number of Chances $N$ $n$ $\nu$, leaves $N$ $n$ $\nu - Y$ $y$
$\upsilon$ the Sum of all the other Seven Products; in all of which one
or more of the three Persons are surviving.

To make this yet more evident, I have added \textit{Fig}.\ 8.\ wherein
these Eight several Products are at one view exhibited.  Let the
rectangular Parallelepipedon on $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ be 
constituted of the sides $A$ $B$, $G$ $H$, \textit{\&amp;c.}\ proportional
to $N$ the number of the younger Age; $A$ $C$, $B$ $D$, \textit{\&amp;c.}\
proportional to $n$; and $A$ $G$, $C$ $E$, \&amp;c. proportional to the
number of the Elder, or $\nu$.  And the whole Parallelepipedon shall be
as the Product $N$ $n$ or our whole number of Chances.  Let $B$ $P$ be
as $R$, and $A$ $P$ as $Y$: let $C$ $L$ be as $r$, and $L$ $n$ as $y$;
and $G$ $N$ as $\varrho$ and $N$ $A$ as $\upsilon$; and let the Plain $P$
$R$ $e$ $a$ be made parallel to the [607] plain $A$ $C$ $G$ $E$; the plain $N$
$V$ $b$ $Y$ parallel to $A$ $B$ $C$ $D$; and the plain $L$ $X$ $T$ $Q$
parallel to the plain $A$ $B$ $G$ $H$.  And our first Product $R$ $r$
$\varrho$ shall be as the Solid $S$ $T$ $W$ $I$ $F$ $Z$ $e$ $b$. The
Second, or $r$ $\varrho$ $Y$ will be as the Solid $E$ $Y$ $Z$ $e$ $Q$ $S$
$M$ $I$. The Third, $R$ $\varrho$ $y$, as the Solid $R$ $H$ $O$ $V$ $W$ $I$
$S$ $T$. And the Fourth, $R$ $r$ $\upsilon$, as the Solid $Z$ $a$ $b$
$D$ $W$ $X$ $I$ $E$. Fifthly, $\varrho$ $Y$ $y$, as the Solid $G$ $Q$ $R$
$S$ $I$ $M$ $N$ $O$. Sixthly, $r$ $y$ $\upsilon$, as $I$ $K$ $L$ $M$ $G$
$Y$ $Z$ $A$.  Seventhly. $R$ $y$ $\upsilon$, as the Solid $I$ $K$ $P$
$O$ $B$ $X$ $V$ $W$. And Lastly, $A$ $I$ $K$ $L$ $M$ $N$ $O$ $P$ will be
as the Product of the 3 numbers of persons dead, or $Y$ $y$ $\upsilon$. 
I shall not apply this in all the cases thereof for brevity sake; only
to shew in one how all the rest may be performed, let it be demanded
what is the value of the Reversion of the younger Life after the two
elder proposed. The proportion is as the whole number of Chances, or $N$
$n$ $\upsilon$ to the Product $R$ $y$ $\upsilon$, so is the certain
present value of the Sum payable after any term proposed, to the value
due to such Chances as the younger person has to bury both the elder, by
the term proposed; which therefore he is to pay for.  Here it is to be
noted, that the first term of all these Proportions is the same
throughout, viz.\ $N$ $n$ $\nu$.  The Second changing yearly according
to the Decrease of $B$, $r$, $\varrho$, and Encrease of $Y$, $y$,
$\upsilon$.  And the third are successively the present values of Money
payable after one, two, three, dc. years, according to the rate of
Interest agreed on. These numbers, which are in all cases of Annuities
of necessary use, I have put into the following Table, they being the
Decimal values of One Pound payable after the number of years in the
Margent, at the rate of 6 \textit{per Cent.} [608] 

\begin{figure}
  \begin{center}
    \epsfig{file=halley_fig_2.eps,width=7cm,width=7cm,clip=}
  \end{center}
\end{figure}

\begin{center}
  \begin{tabular}{|c|c|c|c|c|c|}
\hline
Years. &amp; Present va-   &amp; Years. &amp; Present va-   &amp; Years. &amp; Present va-   \\
       &amp; lue of 1 $l$. &amp;        &amp; lue of 1 $l$. &amp;        &amp; lue of 1 $l$. \\
\hline
\z1    &amp; 0,9434        &amp; 19     &amp; 0,3305        &amp; \z37   &amp; 0,1158 \\
\z2    &amp; 0,8900        &amp; 20     &amp; 0,3118        &amp; \z38   &amp; 0,1092 \\
\z3    &amp; 0,8396        &amp; 21     &amp; 0 2941        &amp; \z39   &amp; 0,1031 \\
\z4    &amp; 0,7921        &amp; 22     &amp; 0,2775        &amp; \z40   &amp; 0,0972 \\
\z5    &amp; 0,7473        &amp; 23     &amp; 0,2018        &amp; \z45   &amp; 0,0726 \\
\z6    &amp; 0,7050        &amp; 24     &amp; 0:2470        &amp; \z50   &amp; 0,0543 \\
\hline
\z7    &amp; 0,6650        &amp; 25     &amp; 0,2330        &amp; \z55   &amp; 0,0406 \\
\z8    &amp; 0,6274        &amp; 26     &amp; 0,2198        &amp; \z60   &amp; 0,0303 \\
\z9    &amp; 0,5919        &amp; 27     &amp; 0,2074        &amp; \z65   &amp; 0,0227 \\
10     &amp; 0,5584        &amp; 28     &amp; 0,1956        &amp; \z70   &amp; 0,0169 \\
11     &amp; 0,5268        &amp; 29     &amp; 0,1845        &amp; \z75   &amp; 0,0126 \\
12     &amp; 0,4970        &amp; 30     &amp; 0,1741        &amp; \z80   &amp; 0,0094 \\
\hline
13     &amp; 0,4688        &amp; 31     &amp; 0,1643        &amp; \z85   &amp; 0,0071 \\
14     &amp; 0,4423        &amp; 32     &amp; 0,1550        &amp; \z90   &amp; 0,0053 \\
15     &amp; 0,4173        &amp; 33     &amp; 0,1462        &amp; \z95   &amp; 0,0039 \\
16     &amp; 0,3936        &amp; 34     &amp; 0,1379        &amp; 100    &amp; 0,0029 \\
17     &amp; 0,3714        &amp; 35     &amp; 0,1301        &amp;        &amp;        \\
18     &amp; 0,3503        &amp; 36     &amp; 0,1227        &amp;        &amp;        \\
\hline
  \end{tabular}
\end{center}

It were needless to advertise, that the great trouble of working so many
Proportions will be very much alleviated by using Logarithms; and that
instead of using $N$ $n$ $\nu - Y$ $y$ $\upsilon$ for the Second Term of
the Proportion in finding the value of Three Lives, it may suffice to
use only $Y$ $y$ $\upsilon$, and then deducting the Fourth Term so
found out of the Third, the Remainder shall be the present value sought;
or all these Fourth Terms being added together, and deducted out of the
value of the certain Annuity for so many Years, will leave the value of
the contingent Annuity upon the Chance of Mortality of all those three
Lives.  For Example; Let there be Three Lives of 10, 30, and 40 years of
Age proposed, and the Proportions will be thus: 

\begin{center}
  As 661 in 531 in 445 or 156190995, or $N$ $n$ $\nu$ \\
  \begin{tabular}{l@{\ }l@{\ }l}
    to 8 in 8 in 9, or 576, &amp; or $Y$ $y$ $\upsilon$ for the first year, 
      &amp; so 0, 9434. to 0,00000348 \\
    to 15 in 10 in 18, or &amp; 4320, for the second year, 
      &amp; so 0,8900. to 0,00002462 \\
    to 21 in 24 in 28, or &amp; 14112 for the third year,    
      &amp; so 0,8396. to 0,00008128 \\
    to 27 in 32 in 38, &amp; for the fourth year,    
      &amp; so 0,7021. to 0,00016650 \\
    to 33 in 41 in 48, &amp; for the fifth year,
      &amp; so 0,7473. to 0,00031071 \\
    to 39 in 50 in 58, &amp; for the sixth year, 
      &amp; so 0,7050. to 0,00051051 
  \end{tabular}
\end{center}
\begin{flushright}
  [609]
\end{flushright}
And so forth to the 60th year, when we suppose the elder Life of Forty
certainly to be expired; from whence till Seventy we must compute for
the First and Second only, and from thence to Ninety for the single
youngest Life.  Then the Sum Total of all these Fourth Proportionals
being taken out of the value of a certain Annuity for 90 Years, being
16, 58 years Purchase, shall leave the just value to be paid for an
Annuity during the whole term of the Lives of three Persons of the Ages
proposed.  And note, that it will not be necessary to compute for every
year singly, but that in most cases every 4th or 5th year may suffice,
interpoling for the intermediate years \textit{secundum artem}.

It may be objected, that the different Salubrity of places does hinder
this Proposal from being \textit{universal}; nor can it be  denied.  But
by the number that die, being 1174.\ \textit{per Annum} in 34000, it
does appear that about a 30th part die yearly, as \textit{Sir William
Petty} has computed for \textit{London}; and the number that die in
Infancy, is a good Argument that the Air is but indifferently
salubrious.  So that by what I can learn, there cannot perhaps be one
better place proposed for a Standard.  At least 'tis desired that in
imitation hereof the Curious in other Cities would attempt something of
the same nature, than which nothing perhaps can be more uselul. [610]

\newpage

\noindent\hangindent=2em
{\Large \textbf{\textit{I. Some further Considerations on the} Breslaw \textit{Bills of
Mortality.  By the same Hand} \&amp;c.}}

\bigskip

SIR, 

\bigskip

What I gave you in my former Discourse on these Bills, was chiefly
designed for the Computation of the Values of Annuities on Lives,
wherein I believe I have performed what the short Period of my
Observations would permit, in relation to exactness, but at the same
time do earnestly desire, that their Learned Author Dr.\ Newman of
Breslaw would please to continue them after the same manner for yet some
years further, that so the casual Irregularities and apparent
Discordance in the Table, \textit{p.\ 599}. may by a certain number of
Chances be rectified and ascertained. 

Were this \textit{Calculus} founded on the Experience of a very great
number of Years, it would be very well worth the while to think of.
Methods for facilitating the Computation of the Value of two, three, or
more Lives; which as proposed in my former, seems (as I am inform'd) a
Work of too much Difficulty for the ordinary Arithmetician to undertake.
 I have sought, if it were possible, to find a Theorem that might be
more concise than the Rules there laid down, but in vain; for all that
can be done to expedite it, is by Tables of Logarithms ready computed,
to exhibit the \textit{Rationes} of $N$ to $Y$ in each single Life, for
every third, fourth or fifth Year of Age, as occasion shall require; and
these Logarithms being added to the Logarithms of the present Value of
Money payable after so many Years, will give a Series of Numbers,  the
Sum of which will show the Value of the Annuity sought. However for each
Number of this Series two Logarithms for a single Life, three for two
Lives, and four for three Lives, must necessarily be added together. If
you think the matter, under the uncertainties I have mentioned, to
deserve it, I shall shortly give you such a Table of Logarithms as I
speak of, and an Example or two of the use thereof: But by Vulgar
Arithmetick the labour of these Numbers were immense; and nothing will
more recommend the useful Invention of Logarithms to all Lovers of
Numbers, than the advantage of Dispatch in this and such like
Computations. 

Besides the uses mentioned in my former, it may perhaps not be an
unacceptable thing to infer from the same Tables, how unjustly we
repine at the shortness of our Lives, and think our selves wronged if we
attain not Old Age; whereas it appears hereby, that the one half of
those that are born are dead in Seventeen years time, 1238 being in that
time reduced to 616.  So that instead of murmuring at what we call an
untimely Death, we ought with Patience and unconcern to submit to that
Dissolution which is the necessary Condition of our perishable
Materials, and of our nice and frail Structure and Composition: And to
account it as a Blessing that we have survived, perhaps by many Years,
that Period of Life, whereat the one half of the whole Race of Mankind
does not arrive. 

A second Observation I make upon the said Table, is that the Growth and
Encrease of Mankind is not so much stinted by any thing in the Nature of
the Species, as it is from the cautious difficulty most People make to
adventure on the state of Marriage, from the prospect of the Trouble and
Charge of providing for a Family.  Nor are the poorer sort of People
herein to be blamed, since their difficulty of subsisting is occasioned
by the unequal Distribution of Possessions, all being necessarily fed
from the Earth, of which yet so few are Masters.  So that besides
themselves and Families, they are yet to work for those who own the
Ground that feeds them:  And of [655] such does by very much the greater part
of Mankind consist; otherwise it is plain, that there might well be four
times as many Births as we now find.  For by computation from the Table,
I find that there are nearly  15000 Persons above 16 and under 45, of
which at least 7000 are Women capable to bear Children.  Of these
notwithstanding there are but 1238 born yearly, which is but little more
than a sixth part.  So that about one in six of these Women do breed
yearly; whereas were they all married, it would not appear strange or
unlikely, that four of six should bring a Child every year.  The
Political Consequences hereof I shall not insist on, only the Strength
and Glory of a King being in the multitude of his Subjects, I shall only
hint, that above all things, Celibacy ought to be discouraged as, bv
extraordinary Taxing and Military Service: And those who have numerous
Families of Children to be countenanced and encouraged by such Laws as
the \textit{Jus trium Liberorum} among the \textit{Romans}. But
especially, by an effectual Care to provide for the Subsistence of the
Poor, by finding them Employments, whereby they may earn their Bread,
without being chargeable to the Publick. [656]

\bigskip\bigskip

\noindent
[\textit{Philosophical Transactions of the Royal Society of London},
\textbf{17} (1693), 596--610 \&amp; 654--656.]

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